Polymer Foams Handbook: Engineering and Biomechanics Applications and Design Guide

Polymer Foams Handbook: Engineering and Biomechanics Applications and Design Guide by Nigel J. Mills • ISBN: 075068069...

Author: Nigel Mills

924 downloads 2426 Views 11MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Polymer Foams Handbook: Engineering and Biomechanics Applications and Design Guide

by Nigel J. Mills

• ISBN: 0750680695 • Publisher: Butterworth-Heinemann • Pub. Date: April 2007

Foreword

This book explores the applications of polymer foams. It attempts to explain their mechanical properties in terms of polymer properties and the foam microstructure. The first chapter introduces geometrical concepts that are used throughout the book. The subsequent three chapters deal separately with the microstructure and processing of polyurethane foams, foamed thermoplastics, and bead-foam mouldings. Their different processing routes and microstructures mean that a combined treatment would be confusing. Surface tension may mainly determine the microstructure of PU foams, but high-stress melt flow is more important for foamed thermoplastics. Concepts in the mechanical property area are introduced in Chapter 5 before Chapter 6 covers finite element analysis (FEA) for the complex geometries of many foam products. Case studies are included for two reasons. Firstly, they are increasingly used to motivate students to study the relevant theory and to understand major industries; a variety of activities such as literature searches and product dismantling can be used as the basis for student presentations. Secondly, specialised foam-based industries tend to remain compartmentalised, but could learn a lot from each other. Thus the areas of foam seating, protective packaging, safety helmets are included; each is associated with one or more theory chapters. There are two strategies for reading this book. One is to read the case studies alone, and use the computer programs to illustrate the foam selection and properties. The other is to read a case study together with the appropriate background theory on the mechanics and materials science. If the reader’s background is weak in polymer materials science, it is recommended that he/she should read a general textbook, such as the authors Plastics. This book is intended to compliment previous books with different approaches. Mustin (1968)1 viewed foam packaging from the military engineering viewpoint; how to design packaging so that supplies could survive air-drops. Hilyard’s (1982)2 multiauthor book reviewed many areas of mechanical properties, but concentrated on polyurethane systems. Gibson and Ashby’s (1988)3 book, which surveyed all cellular solids, contains many interesting ideas. However it gives the impression that the mechanical properties of foams are fully explained. It uses a dimensional approach to avoid full analyses of deformation, which are now available. Some of the proposed deformation mechanisms are less important than suggested, and some are not observed in polymer foams. Hilyard and Cunningham’s (1994)4 book contains a good review of the micromechanics of foam elasticity by Kraynik and Warren, and useful chapters on the mechanisms of heat transfer and gas diffusion. Finally Klempner and Sendijarevic (2004)5 book reviews the chemistry and processing of all the major polymer families. This book covers the principles which provide a framework for foam developments. It has become easy to search and access literature electronically, but the user should be aware of the advantage and shortcomings of databases such as Google

xviii Foreword Scholar, Science Direct, and RAPRA databases in keeping their coverage of the foam literature up to date. Writing this book has been a voyage of discovery. I am grateful to the efforts of collaborators such as Adam Gilchrist and Miguel Rodriguez-Perez, and to PhD students, in particular Hanzing Zhu, Stephanie Ankrah, Raquel Verdejo, Yago Masso-Moreu, Iona Lyn, and Catherine Fitzgerald. Nigel Mills September 2006

Notes 1. Mustin G.S. (1968) Theory and Practise of Cushion Design, Shock and Vibration Information Center, US Department of Defence. 2. Hilyard N.C. Ed. (1982) Mechanics of Cellular Plastics, Macmillan, London. 3. Gibson L.J. & Ashby M.F. (1988) Cellular Solids, Pergamon, Oxford. (2nd Edn., Cambridge University Press, Oxford, 1997). 4. Hilyard N.C. & Cunningham A. Eds. (1994) Low Density Cellular Plastics, Chapman and Hall, London. 5. Klempner D. & Sendijarevic V. Eds. (2004) Handbook of Polymeric Foams and Foam Technology, 2nd Edn., Hanser, Munich.

Acknowledgements

Figures 1.2a and 7.17 Reproduced from Youssef S. et al. (2005) Finite element modelling of the actual structure of cellular materials determined by X-ray tomography, Acta Mater. 53, 719–730, with permission from Elsevier Ltd. Figure 1.2b Reproduced from Kraynik et al. (1999) ‘Foam micromechanics’ p. 274, Figure 10, in Foams and Emulsions, Eds. Sadoc J.F. & Rivier N., Nato ASI Series E, Vol. 354, with permission from Kluwer. Figure 1.2c Reproduced from Kusner R. & Sullivan J.M. (1966) Comparing the Weaire–Phelan equal-volume foam to Kelvin’s foam, of Forma, 11, 233–242, with permission from Scipress, Toyko. Figure 1.2d Reproduced from Weaire et al. (1999) The structure and geometry of foams’, in Foams and Emulsions, Eds. Sadoc J.F. & Rivier N., Nato ASI Series E, Vol. 354, with permission from Kluwer. Figure 1.3 Reproduced from Gong L. et al. (2005) Compressive response of opencell foams. Part 1. Morphology and elastic properties, Int. J. Solids Struct. 42, 1355, with permission from Elsevier Ltd. Figures 1.7 and 7.4 Reproduced from Elliott J.A. et al. (2002) In-situ deformation of an open-cell flexible polyurethane foam characterised by 3D computed tomography, J. Mater. Sci. 37, 1547–1555, with permission from Kluwer. Figure 1.8 Reproduced from Schulmeister V. (1998) Modelling of the mechanical properties of low-density foams, Ph.D. thesis, Technical University of Delft, with permission from Shaker Publishing, Maastricht. Figures 1.11 and 1.12 Reproduced from Kraynik A.D. et al. (2003) Structure of monodisperse foam, Phys. Rev. E 67, with permission from The American Physical Society. Figures 2.1 and 2.2 Reproduced from Herrington R. & Hock K. (1997) Dow polyurethanes: flexible foams (2nd edn), with permission from Dow Chemical Company. Figure 2.3 Reproduced from Neff R. & Macosko C.W. (1995) A model for modulus development in flexible polyurethane foam, Proc. Polyurethanes September, 344–352, with permission from Society of the Plastics Industry. Figure 2.4 Reproduced from Klemperer D. & Frisch, K. Eds. (2001) Advances in urethane science and technology, chapter 2 by A. Weier and G. Burkhardt, with permission from Rapra.

xx Acknowledgements Figures 2.4b and 2.12 Reproduced from Yasunaga K. et al. (1996) Study of cell opening in flexible polyurethane foam, J. Cell. Plast. 32, 427–448, with permission from Sage Publications. Figure 2.6 Reproduced from Kaushiva B.D. & Wilkes G.L. (2000) Alteration of polyurea hard domain morphology by diethanol amine in molded polyurethane foams, Polymer 41, 6981–6986, with permission from Elsevier Ltd. Figure 2.8 Reproduced with permission from Dow Chemical Company. Figure 2.9 Reproduced from Van der Heide E., van Asselen O.L.J. et al. (1999) Tensile deformation behaviour of the polymer phase of flexible polyurethane foams and polyurethane elastomers, Macromol. Symp. 147, 127–137, with permission from Wiley & Sons Inc. Figure 2.10 Reproduced from Qi H.J. & Boyce M.C. (2005) Stress–strain behaviour of thermoplastic polyurethanes, Mech. Matl. 37, 817–839, with permission from Elsevier Ltd. Figure 2.11 Reproduced from Van der Schuur M. et al. (2004) Elastic behaviour of flexible polyether(urethane–urea) foam materials, Polymer 45, 2721–2727, with permission from Elsevier Ltd. Figure 2.13 Reproduced from Kleiner G.A. et al. (1984) J. Cell. Plast. 49–57, with permission from Sage Publications. Figure 2.14 Reproduced with permission from Dow Chemical Company. Figure 3.2 Reproduced from Guan and Pitchumani (2007) A micromechanical model for the elastic properties of semicrystalline thermoplastic polymers, Poly. Eng. Sci. 44, 433, with permission from Wiley & Sons Inc. Figures 3.5 and 3.6 Reproduced from Ruinaard H. (2006) Elongational viscosity as a tool to predict the foamability of polyolefins, J. Cell. Plast. 42, 207–220, with permission from Society of Plastic Engineers. Figure 3.9b Reproduced from Koopmans R.J. et al. (2000) Modelling foam growth in thermoplastics, Adv. Mater. 12, 1873, with permission from Wiley & Sons Inc. Figure 3.10 Reproduced from Everitt S.L. et al. (2006) Competition and interaction of polydisperse bubbles in polymer foams, J. Non-Newt. Fluid Mech., with permission from Elsevier Ltd. Figure 3.15 Reproduced from Rodriguez-Perez M.A. et al. (2005) Characterisation of the matrix polymer morphology of polyolefin foams by Raman spectroscopy, Polymer 46, 12093–12102, with permission from Elsevier Ltd. Figures 3.16 and 3.17 Reproduced from Almanza O, Rodriguez-Perez M.A. et al. (2005) Comparative study on the lamellar structure of polyethylene foams, Euro. Poly. J. 41, 599–609, with permission from Elsevier Ltd. Figure 4.1b Reproduced from Cigna G. et al. (1986) Morphological and kinetic study of EPS pre-expansion and effects on foam properties, Cell. Polym. 5, 241–268, with permission from Rapra.

Acknowledgements

xxi

Figures 4.1a and 4.2 Reproduced from Fen-Chong T. et al. (1999b) Viscoelastic characteristics of pentane-swollen polystyrene beads, J. Appl. Polym. Sci. 73, 2463–2472, with permission from J Wiley & Sons Inc. Figure 4.3a Reproduced from Tuladhar T.R. & Mackley M.R. (2004) Experimental observations and modelling related to foaming and bubble growth from pentane loaded polystyrene melts, Chem. Eng. Sci. 59, 5997–6014, with permission from Elsevier Ltd. Figure 4.7 Reproduced from Rossacci J. & Shivkumar S. (2003) J. Mater. Sci. 38, 201, with permission from Kluwer. Figures 4.9 and 4.10 Reproduced from Jarvela P. et al. (1986a) A method to measure the fusion strength of EPS beads, J. Mater. Sci. 21, 3139, with permission from Kluwer. Figure 5.7 Reproduced from Wada A. et al. (2003) A method to measure shearing modulus of the foamed core for sandwich plates, Compos. Struct. 60, 385–390, with permission from Elsevier Ltd. Figures 6.13 and 6.14 Reproduced from Deshpande V.S. & Fleck N. (2001) Multiaxial yield behaviour of polymer foams, Acta. Mater. 49, 1859–1866, with permission from Elsevier Ltd. Figure 7.10 Reproduced from Laroussi M. et al. (2002) Foam mechanics: nonlinear response of an elastic 3D-periodic microstructure, Int. J. Solids Str. 39, 3599–3623, with permission from Elsevier Ltd. Figure 7.16 Reproduced from Zhu H.X. & Windle A.H. (2002) Effects of cell irregularity on the high strain compression of open-cell foams, Acta Mater. 50, 1041–1052, with permission from Elsevier Ltd. Figure 7.18 Reproduced from Gong L. et al. (2005b) On the stability of Kelvin cell foams under compressive loads, J. Mech. Phys. Solids 53, 771–794, with permission from Elsevier Ltd. Figures 8.1a and 8.17 Reproduced from Cummings A. & Beadle S.P. (1993) Acoustic properties of reticulated plastic foams, J. Sound Vibr. 175, 115, with permission from Academic Press. Figure 8.3 Reproduced from Hilyard N.C. & Collier P. (1987) A structural model for air-flow in flexible PU foams, Cell. Polym. 6, 9–26, with permission from Rapra. Figure 9.1 Reproduced from Hubbard R.P. et al. (1993) New biomechanical models for automotive seat design, SAE SP-963, Seat System Comfort and Safety, pp. 35–42, with permission from SAE. Figure 9.4 Reproduced from Setyabudky R.F., Ali A. et al. (1997) Measuring and modelling of human soft tissue and seat interaction, SAE SP-1242, Progress with Human Factors in Automotive Design, pp. 135–142, with permission from SAE. Figure 9.6b Reproduced from Kinkelaar M.R. et al. (1998) Vibrational characterization of various polyurethane foams employed in automotive seating applications, J. Cell. Plast. 34, 155–173, with permission from Sage Publications.

xxii Acknowledgements Figure 9.11 Reproduced from Cavender K.D. & Kinkelaar M.R. (1996), Real time dynamic comfort and performance factors of polyurethane foam in automotive seating, SP-1155 SAE, Automotive Design Advancements in Human Factors, with permission from SAE. Figure 9.13 Reproduced from Shen W. & Vertiz A.M. (1997) Redefining seat comfort, SAE SP-1242, Progress in Human Factors in Automotive Design, pp. 161–168, with permission from SAE. Figure 9.14 Reproduced from Bouten C.V. et al. (2003) The etiology of pressure ulcers: skin deep or muscle bound? Arch. Phys. Med. Rehabil. 84, 616–619, with permission from American Congress of Rehabilitation Medicine. Figure 9.15 Reproduced from Verver M.M., van Hoof J. et al. (2004) A FE model of the human buttocks for prediction of seat pressure distributions, Comp. Meth. Biomech. Biomed. Eng. 7, 193, with permission from Taylor & Francis. Figure 9.16 Reproduced from Broos R. et al. (2001) Endurance of PU automotive seating foam under various temperature and heating conditions, Cell. Polym. 19, 169–204, with permission from Rapra. Figure 11.15b Reproduced with permission of McKown S. Figure 11.17 Reproduced from Viot P. & Bernard D. (2005) Impact test deformations of polypropylene foam samples followed by microtomography, J. Mater. Sci. 41, 1277–1279, with permission from Kluwer. Figure 12.3 Reproduced from Sek M.A. & Kirkpatrick J. (1997) Prediction of the cushioning properties of corrugated cardboard. Pack. Sci. Tech. 10, 91, Figure 6, Comparison of simulated and experimental cushioning curves, with permission from Wiley & Sons Inc. Figure 12.4 Reproduced with permission from Zotefoams Brochure for LD24 cushion curve. Figure 12.15 Reproduced from Totten T.L. et al. (1990) The effects of multiple impacts on the cushioning properties of closed cell foams, Pack. Tech. Sci. 3, 117–122, with permission from Wiley & Sons Inc. Figure 13.13 Adapted from Misevich K.W. & Cavanagh P.R. (1984) Material aspects of modeling shoe/foot interaction, p. 58 Figure 10, in Sports Shoes and Playing Surfaces, Frederick E.C. Ed. Human Kinetics Publishers Inc. © Nike. Figure 13.16 Reproduced from Kinoshita H. & Bates B.T. (1996) The effect of environmental temperature on the properties of running shoes, J. Appl. Biomech. 12, 258–268, with permission from Human Kinetics Publishers Inc. Figure 14.3 Reproduced from Smith T.A. et al. (1994) Evaluation and replication of impact damage to bicycle helmets, Accid. Anal. Prev. 26, 795–802, with permission from Elsevier Ltd. Figure 14.4 Reproduced from L.B. Larsen et al. (1991) Epidemiology of bicyclist’s injuries, Conference IRCOBI, 217–230, with permission from IRCOBI.

Acknowledgements

xxiii

Figure 15.1 Reproduced from Stupak P.R. et al. (1991) The effect of bead fusion on the energy absorption of polystyrene foam, J. Cell. Plast. 27, with permission from Sage Publications. Figure 15.12 Reproduced from Choi S. & Sankar B.V. (2005) A micromechanical method to predict the fracture toughness of cellular materials, Int. J. Solids Struct. 42, 1797, with permission from Elsevier Ltd. Figure 15.14 Reproduced from McIntyre A. & Anderson G.E. (1979) Fracture properties of a rigid PU foam over a range of densities, Polymer 20, 247–253, with permission from Elsevier Ltd. Figure 15.15 Reproduced from Zenkert D. & Bäcklund J. (1989) PVC sandwich core materials: mode I fracture toughness, Comp. Sci. Tech. 34, 225–242, with permission from Elsevier Ltd. Figures 15.16 and 18.2 Reproduced from Danielsson M. (1996) Toughened rigid foam core material for use in sandwich construction, Cell. Polym. 15, 417–435, with permission from RAPRA. Figure 15.17 Reproduced from Stupak P.R. et al. (1991) The effect of bead fusion on the energy absorption of polystyrene foam. Part 1. Fracture toughness, J. Cell. Plast. 27, 484, with permission from Sage Publications. Figure 16.15 Reproduced from Kostopoulos V. et al. (2002) FEA of impact damage response of composite motorcycle safety helmets, Composites Part B 33, 97–107, with permission from Elsevier Ltd. Figure 17.1 Reproduced from Okuizumi H., Harada A. et al. (1998) Effect on the femur of a new hip fracture preventative system using dropped-weight impact testing, J. Bone Miner. Res. 13, 1940–1945, with permission from American Society for Bone and Mineral Research. Figure 17.2a Reproduced with permission from Dr Jones, Imperial College, London, UK. Figure 17.2b Reproduced from Bousson V. et al. (2004) Cortical bone in the human femoral neck: 3D appearance and porosity using synchrotron radiation, J. Bone Min. Res. 19, 794–801, with permission from American Society for Bone and Mineral Research. Figure 17.5 Reproduced from Robinovitch S.N. et al. (1994) Force attenuation in trochanteric soft tissues during impact from a fall, J. Orthopaed. Res. 13, 956–962, with permission from Orthopaedic Research Society. Figure 17.6 Redrawn from Askegaard V. & Lauritzen J.B. (1995) Load on the hip in a stiff sideways fall, Eur. J. Muscoloskel. Res. 4, 111–115, Scandinavian University Press. Figures 17.7 and 21.2 Reproduced from Ellis H. et al. (1991) Human CrossSectional Anatomy, with permission from Elsevier Ltd.

xxiv Acknowledgements Figures 17.9 and 17.11 Reproduced from Robinovitch S.N. et al. (1995) Energy shunting hip padding system attenuates femoral impact force in a simulated fall, J. Biomech. Eng. Trans ASME, 117, 409–413, with permission from ASME. Figures 17.9b and 17.12a Reproduced from Derler S. et al. (2005) Anatomical hip model for the mechanical testing of hip protectors, Med. Eng. Phys. 27, 475, with permission from Elsevier Ltd. Figure 17.12b Reproduced from Van Schoor N.M., van der Veen A.J. et al. (2006) Biomechanical comparison of hard and soft hip protectors, and the influence of soft tissue, Bone 39, 401–407, with permission from Elsevier Ltd. Figure 18.3 Reproduced from Thomas T., Mahfuz H. et al. (2004) High strain rate response of crosslinked and linear PVC cores, J. Reinf. Plast. Compos. 23, 739–749, with permission from Sage Publications. Figure 18.4a Reproduced from Microstructures of H130 and H130 PVC foams (Kanny K., Mahfuz H. et al. (2004) Static and dynamic characterization of polymer foams under shear loads, J. Comp. Matl. 38, 629), with permission from Sage Publications. Figure 18.4b Redrawn from Gundberg T. (2002) Foam cores in the marine industry: www.boatdesign.net Figures 18.10 and 18.11 Reproduced from Kim J. & Swanson S.R. (2001) Design of sandwich structures for concentrated loading, Comp. Str. 52, 365–373, with permission from Elsevier Ltd. Figure 18.13 Reproduced from Magnitude of residual dent vs. indentation magnitude for H 60 PVC foam core (Zenkert D., Shipska A. & Persson K. (2004) Static indentation and unloading response of sandwich beams, Composites B 35, 511), with permission from Elsevier Ltd. Figure 18.14 Adapted from Rizov V., Shipsha A. & Zenkert D. (2005) Indentation study of core sandwich composite panels, Compos. Struct. 69, 95–102. Elsevier Ltd. Figure 18.15 Reproduced from Eeckhaut G. & Cunningham A. (1996) The elimination of radiative heat transfer in fine celled PU rigid foams, J. Cell. Plast. 32, 528–552. Figure 18.16 Reproduced from Ahern A. et al. (2005) The conductivity of foams: a generalisation of the electrical to the thermal case, Coll. Surf. 263, 275–279, with permission from Elsevier Ltd. Figure 18.17 Reproduced from Quenard D. et al. (1998) Heat transfer in the packing of cellular pellets: microstructure and apparent thermal conductivity, 14th ECTP Proceedings, 1089–1095, with permission from Pion Ltd. Figure 18.18 Reproduced from Wilkes K.E., Yarborough D.W. et al. (2002) Aging of polyurethane foam insulation in simulated refrigerator panels- 3-year results with third generation blowing agents, with permission from Alliance for the Polyurethanes Industry.

Acknowledgements

xxv

Figures 19.17 and 19.18 Reproduced from Rodriguez-Perez M.A., Ruiz-Herrero J.L. et al. (2006) Gas diffusion in polyolefin foams during creep tests, Cell. Polym. 25, 221–236, with permission from Rapra. Figure 20.3 Reproduced from Shafee E.E. & Naguib H.F. (2003) Water sorption in cross-linked poly(vinyl alcohol) networks, Polymer 44, 1647–1653, with permission from Elsevier Ltd. Figure 20.4 Reproduced from www.foamex.com/technical/resanddel.asp SIP felt oil wicking as a function of firmness with permission from Foamex. Figure 20.5a Reproduced from Chen J., Park H., Park K. (1999) Synthesis of superporous hydrogels: hydrogels with fast swelling and superabsorbent properties, J. Biomed. Res. 44, 53–62, with permission from Wiley & Sons Inc. Figure 20.5b Reproduced from Braun J. et al. (2003) Non-destructive, 3D monitoring of water absorption in PU foams using MRI, Poly. Test 22, 761–767, with permission from Elsevier Ltd. Figure 20.10 Reproduced from Dusˇkov M. (1997) Materials research on EPS 20 and EPS15 under representative conditions in pavement structures, Geotext. Geomembr. 15, 147–181, with permission from Elsevier Ltd. Figure 20.11 Reproduced from Gnip I.Y., Kersulis V. et al. (2006) Water absorption of expanded polystyrene boards, Poly. Test. 25, 635–641, with permission from Elsevier Ltd. Figure 20.13 Reproduced from Ojanen T. & Kokko E. (1997) Moisture performance analysis of EPS frost insulation, in STP 1320 Insulation Materials, Testing and Applications, pp. 442–455, with permission from ASTM. Figure 20.14 Reproduced from Dement’ev A.G. et al. (1996) Diffusion and sorption of water vapour in polyurethane foam, Polym. Sci. USSR 31, 2291–2296, with permission from Elsevier Ltd. Figure 20.15 Reproduced from Mondal P. & Khakhar D.V. (2004) Regulation of cell structure in water-blown rigid polyurethane foam, Macromo. Symp. 216, 241–254, with permission from Iupac. Figure 21.19a Reproduced from McIntosh A., McCrory P. & Finch C.F. (2004) Performance enhanced headgear: a scientific approach to the development of protective headgear, Br. J. Sports Med. 38, 46–49, with permission from BMJ Ltd.

Chapter 1

Introduction to polymer foam microstructure

Chapter contents 1.1 Open- and closed-cell foams 1.2 Relative density: wet and dry foams 1.3 Edges 1.4 Vertices 1.5 Faces 1.6 Cell geometry 1.7 Cells 1.8 Foam microstructural models 1.9 Bead foams References

2 3 5 6 7 8 9 12 17 17

2 Polymer Foams Handbook This chapter introduces concepts, which will be used in Chapters 2–4 to describe the geometry of the three main types of foams. These concepts come from scientific fields such as physics, topology, the study of soap froths, and that of plant structures. Microstructural models, which will be used later to explain mechanical properties, will be introduced, since these affect our view of the microstructures. Background on polymer structure property relationships is provided by Mills (2005), while the applications of foams are introduced at foamstudies.bham.ac.uk

1.1

Open- and closed-cell foams Figure 1.1 compares typical microstructures of open- and closed-cell polymer foams, respectively, as seen in a scanning electron microscope (SEM). The good depth of focus allows complete cells to be seen in the interior of open-cell polyurethane (PU) foams. Air can

Cut edge Edge Vertex

(a)

Cut face Cut edge Face Vertex Edge

(b)

Figure 1.1

SEM photograph of (a) PU open-cell foam of density 28 kg m⫺3; and (b) closed-cell low density polyethylene (LDPE) foam of density 24 kg m⫺3.

Chapter 1 Introduction to polymer foam microstructure

3

pass freely between the cells of such foams. Although in Figure 1.1a all the cell faces are open, only a small fraction of cell faces need to be open to create continuous air passages. In a typical closed-cell foamed thermoplastic, each cell is surrounded by connected faces. Partial cells, with cut faces and edges, are visible on the cut surfaces (Fig. 1.1b), while complete cells exist in the interior of the sample. The cell faces, although thicker and stronger than those in closed-cell PU foams, can sometimes be split or otherwise damaged.

1.2

Relative density: wet and dry foams The foam relative density R is defined as R⫽

ρ

f

ρp

(1.1)

where ρf is the foam density and ρp is the polymer density, which is generally in the range 900–1200 kg m⫺3. When no other phases (such as glass fibres or solid fire-retardant additives) are present, R is the volume fraction of polymer in the foam. Ceramicists prefer the concept of porosity, which is defined as 1 ⫺ R. Low-density foams have R ⬍ 0.1, while structural foam injection mouldings have R in the range 0.4–0.8. Since their microstructure and properties (Throne, 1996) are closer to bulk polymers than to low-density foams, they will not be considered here. Figure 1.2 shows stages in the development of foams: (a) Isolated spherical bubbles grow in the liquid polymer. Spheres have the minimum surface area for a given volume, so the surface energy of the gas–liquid interface is minimised. (b) When bubbles touch, their shapes distort. Equal size bubbles, packed in a face centred cubic (FCC) array, would touch when R ⫽ 0.26. Bubbles with a distribution of sizes, pack to a slightly higher density before they touch. In this wet closed-cell foam, thin, planar faces occur between the cells. Curved surfaces enclose liquid in the cell edges and vertices. The term wet was coined for soap froths (Weaire and Hutzler, 1999), implying a significant water content. If the foam rises, while being constrained in width, the cell shapes to become anisotropic. (c) In the limit as the foam relative density R : 0, the closed-cell foam becomes dry. This stage is never reached in polymer foams, but is a useful idealisation for modelling. When water drains from soap froths under gravity, the cell faces are stabilised by a bilayer of surfactant molecules, and the edges are of the same thickness as the faces.

4 Polymer Foams Handbook

500 µm (a)

(b)

(c)

(d)

Figure 1.2

Stages in the development of foams: (a) isolated spherical bubbles (Youssef et al., 2005 © Elsevier), (b) a cell in a wet foam (Kraynik et al., 1999 © Kluwer), (c) a dry Weaire–Phelan foam (Kusner and Sullivan, 1996 ©), and (d) an open-cell foam (Weaire et al., 1999 © Kluwer).

(d) When the faces in a wet closed-cell foam collapse, an open-cell foam is formed. The polymerisation and crosslinking of the PU stabilise the edges and vertices. (e) In soap froths and low viscosity polymers, the foam can collapse back to a liquid. We will see in Chapter 2 that PU foams can be open- or closed-cell, whereas Chapter 3 reveals the great majority of foamed thermoplastics to be closed-cell. Foam structures contain three main elements.


(a)

5

0.5 mm r b

(b)

0.5 mm

Figure 1.3

1.3

(c)

(a) Edge from a large cell size PU foam with density circa 30 kg m⫺3, (b) sections of the edge (Gong et al., 2005 © Elsevier), and (c) idealised cross-section of a Plateau border.

Edges Cell edges are usually straight in unloaded foams. In dry closed-cell foams, the edges have shrunk to lines; if the surface energy is minimised, three faces meet at each edge, with interface angle of 120°. In open-cell PU foams the Plateau border edges have three cusps. The angles between these cusps, seen along the edge, are close to 120° (Fig. 1.1a). The edges are usually relatively stubby, with lengths being only a small multiple of their width. The variation in edge width was characterised by Gong et al. (2005); there is a minimum value midway between vertices (Fig. 1.3). Edges are sometimes incorrectly described as struts; this implies that their main mechanical role is to resist axial compression, which is rarely the case. Plateau (1873) described the shape of edges in soap froths, in which gravitational forces are negligible compared with the surface tension of the water–air interface (a constant). Minimisation of froth surface energy, hence minimisation of its surface area, determines the equilibrium shape of the liquid–gas interface. Since the viscosity of water is low (1.5 ⫻ 10⫺ 3 Ns m⫺2), a soap froth achieves its equilibrium geometry almost instantaneously. The cross-section of a Plateau border consists of three, touching, circular arcs. Smith (1948) considered the shape of a second phase existing at the boundaries between three grains in a metal. He showed that the cusp angle θ at the corners, where the circular arcs met, was given by ⎛ γ ⎞⎟ ⎜ gb ⎟ θ ⫽ 2cos⫺1 ⎜⎜ ⎟ ⎜⎜⎝ γαβ ⎟⎟⎠

(1.2)

6 Polymer Foams Handbook where γgb and γαβ are the surface energies of the grain boundary and the phase boundary, respectively. For open-cell foams γgb becomes the surface energy of the faces prior to collapse, and γαβ the surface energy between the edge polymer and air. These two quantities should be equal, so θ should be 0°, as in a Plateau border. Open-cell PU foams have cusp angles θ ⬍ 10° (Chan and Nakamura, 1969). The edge cross-sectional area A is related to the edge breadth b by ⎛ π⎞ A ⫽ b2 ⎜⎜ 3 ⫺ ⎟⎟⎟ ⎜⎝ 2 ⎟⎠

(1.3)

where b is also the radius of curvature of the interface. When sections through the edges of closed-cell foams are examined, it is easier to measure the radial distance r from the edge midline to the nearest point on the circular arc, than the edge breadth directly, because the boundary between the face and edge may be unclear; r is related to A by A ⫽ 3r 2

1.4

3 ⫺ π/2 (2 ⫺ 3)2

⫽ 6.737 r 2

(1.4)

Vertices Vertices connect edges in a similar way to cast metal nodes connecting tubes in a space–frame structure. Ideally, four cells and four edges meet at each vertex. Figure 1.4 shows a vertex and four half edges in a Kelvin foam, computed with the Surface Evolver software, available free from the University of Minnesota at www.geom.umn.edu/ software/evolver/. The process starts with a dry Kelvin closed-cell foam. A command file, wetfoam.cmd, with an edge spread parameter (S), creates edges of constant isosceles-triangle cross-section; one side has length S/2 and the others are 兹(3/4) S/2. The value of S determines the foam relative density. The edge surfaces are tiled with a small number of triangular facets. A series of steps moves the locations of the facets to reduce the surface area of the foam, refines the triangular faces by dividing them into four, and calculates the minimum surface energy. Phelan et al. (1996) calculated the total volume V per edge, as a function of the distance L between vertex-centres and the edge breadth b at mid-length, in such a foam, as V ⫽

(

)

3 ⫺ π Ⲑ 2 b2 ( L ⫹ 1.50b )

for the case where L ⬎ b.

(1.5)


Figure 1.4

7

A vertex plus half of four edges, modelled using Surface Evolver software, with R ⫽ 0.0276 (S ⫽ 0.2).

Low-density polymer foams contain some higher order vertices; if two fourfold vertices merge along one edge, the resulting vertex connects six edges. Their relative proportion is greater in denser foams.

1.5

Faces In PU foams the face centres are thinner than the outer regions, and the thickness is typically ⬍1 µm. They behave as thin membranes, wrinkling under in-plane compressive forces. Rhodes (1994) used optical reflection microscopy to obtain interference fringes from the faces, hence the thickness distribution. Even here, the thinnest part is significantly thicker than a soap film face, which can be the length of two surfactant molecules. Faces are usually planar in undeformed polymer foams, since there is no pressure difference between the cells. SEM of some sectioned closed-cell thermoplastic foams shows that the face centres are thinner than the outer regions (Fig. 1.5). However, there are no published thickness profiles. The face thicknesses are much greater than in PU foams, since the highly viscous polymer melt resists the effects of the extensional flow (Section 3.3.2).


1 mm

Figure 1.5

1.6

SEM of Zotefoam high density polyethylene (HDPE) of density 98 kg m⫺3 with higher order vertices arrowed, showing the variation of face thickness (Mills, unpublished).

Cell geometry Low-density, closed-cell polymer foams have polygonal cells of variable shape and size. Statistical distributions of the number of faces per cell (f ) and the number of edges per face (n) characterise the foam. In a dry closed-cell foam each cell is a polyhedron, so its geometry must obey rules related to the space-filling packing of polyhedra. This includes Euler’s formula, for any convex three-dimensional (3-D) polyhedron with E edges, F faces, and V vertices V⫺E⫹F⫽2

(1.6)

As four edges meet at every foam vertex, in the polyhedral cell cut from the foam by removing one edge from every vertex, three edges meet at each vertex. As each edge connects two vertices, to allow for double counting, E ⫽ 1.5V. Substituting for V in equation (1.6) gives 3F ⫺ E ⫽ 6

(1.7) – The faces are n-gons, with a mean number of n edges per face. As each edge of the polyhedral cell separates two faces, the total number of – F/2. Substituting for E in equation (1.7) leads to edges in the cell E ⫽ n ⎛ 2⎞ n ⫽ 6 ⎜⎜1 ⫺ ⎟⎟⎟ ⎜⎝ F ⎟⎠

(1.8)


Figure 1.6

9

CT micrograph of Confor 47 slow-recovery PU foam (kindly provided by Skyscan).

– When averaging for all the cells in the foam, F is replaced by f, the mean number of faces per cell, so ⎛ 2⎞ n ⫽ 6 ⎜⎜⎜1 ⫺ ⎟⎟⎟ ⎜⎝ f ⎟⎠

(1.9)

– – ⫽ 5.1. Although penWe see later that in PU foams f 艑 13.0, so n tagonal faces are relatively common, there is a range of face types.

1.7

Cells X-ray micro-tomography, with a resolution of 8 µm, can reveal the structure deep inside a foam, either as a series of slices which can be characterised by image analysis, or as a reconstructed 3-D image. The cell edges in PU foams, usually being thicker than 10 µm, are resolved (Fig. 1.6). Elliott et al. (2002) skeletonised (removed surface pixels until lines remained) the foam structure from a CT scan of a PU foam of density 19 kg m⫺3, and determined the edge length and cell diameter distributions (Fig. 1.7). Montminy et al. (2001; 2004) emphasised that the face shaped distribution (Table 1.1) for PU foams differed from that of soap froths (Matzke, 1946); pentagonal faces were more common

10 Polymer Foams Handbook 350

Number of struts

300 250 200 150 100 50 0 0

160

320

480

640

800

Strut length (µm)

(a) 30

Number of cells

25 20 15 10 5 0 0

502

Figure 1.7

1030

1558

2086

2614

Cell diameter (µm)

(b)

Edge length and cell diameter distributions for a PU sample shown in Figure 6.4 (Elliott et al., 2002 © Kluwer).

Table 1.1 Statistics of polygonal cell faces Face (%) Foam type No. of cells Triangles Quadrilaterals Pentagons Soap froth PU foam

600 106

0 1

11 24

67 55

Hexagons Heptagons⫹ 22 19

⬍1 1

in soap froths, but square faces less common, than in the PU foams. They attributed this to the different viscosities in the foaming processes. The distribution of inter-edge angles was centred on 107° with a standard deviation of 18°. They suggested that the mean angle


Table 1.2

11

Average cell parameters Author

Material

Williams Montminy Voronoi Kraynik

Soap froth, etc. PU Model Model

– f

–n

– θ

14 13.0 14.3 13.7

5.1 4.95

109°28⬘ 107 ⫾ 0.5°

Normal to extrusion direction

Normal to rise direction

x1

x3

Normal to transverse direction

x1

5 mm x2

Figure 1.8

x3

x2

Three sections of an embedded PU foam, showing the cell-shape anisotropy (Shulmeister, 1998 © Shaker Publishing). The sectioned edges can be joined ‘dot-to-dot’ to locate the sectioned faces.

differed from the tetrahedral 109.5° angle either because of the foam having a rise ratio (typically 1.2) or because a few vertices connect more than four edges. NMR microscopy is an alternative characterisation method (Szayna and Voelkel, 1999). Williams (1968) reviewed the cell geometries of collections of soap bubbles, plant cells, or metal grains, all of which tend to have non-uniform size cells. His average parameters are shown in Table 1.2. He stated that the data tends to conform to three rules: 1. the average number of faces per cell f is 14; – is 5.14; 2. the average number of side per face n 3. the vertices are tetrahedral, with inter-edge angles θ close to 109°28⬘. One important parameter is the average axial ratio of the cells; in PU foams the rise of the sheet of liquid foam, while it lies on a belt, leads to anisotropic cell shapes (Fig. 1.8), with a typical axial ratio of 1.2 (Shulmeister, 1998). The average cell size, although frequently measured, is of little importance for mechanical properties. However, it is important for air flow. The range of cell sizes may affect the shape of the stress–strain

12 Polymer Foams Handbook curve of open-cell foams. The mean cell size affects gas diffusivity in closed-cell foams, and therefore affects a mechanism of high strain creep. It affects the gas flow resistance of open-cell foams. Cell size has only a minor influence on heat transfer in closed-cell foams, as it is usually sufficiently small for isothermal conditions to apply. It is important for process quality control, since uncontrolled face fracture leads eventually to the collapse of the foam. There is no parallel with the grain size in metallic systems, which strongly influences the mechanical strength. In contrast, the foam relative density strongly affects all the foam mechanical properties.

1.8

Foam microstructural models Models can either be an approximation to or a representative part of a real foam structure. The former tends to be regular lattice models, while the latter are irregular.

1.8.1 Lattice micromechanics models Kraynik and Warren (1994) reviewed regular lattice models for foams. A regular lattice of pentagonal dodecahedrons nearly matches the average number of sides per face in polymer foams, but it is difficult to analyse. Some models have drawbacks; the hexagonal close packed (HCP) lattice of cells has continuous straight edges running through the microstructure (Ko, 1965), causing unacceptably high Young’s moduli in these directions and high elastic anisotropy. The HCP and the FCC lattices have sixfold vertices, hence do not obey Williams’ (1968) geometric rules. A lattice of tetrakaidecahedra satisfies most of the geometric rules. Thomson (Lord Kelvin, 1887) proposed it as the foam with a minimum surface area. It has 8 non-planar hexagonal surfaces and 6 planar quadrilateral surfaces, all with curved edges, while the angles between edges at vertices are 109.5°. Williams (1968) proposed a β–tetrakaidecahedron, with 2 square faces, 8 pentagonal faces, and 4 hexagonal faces, which has a statistical distribution of face shapes closer to that found in nature or in metal grains, but this has been used little. For modelling (Chapters 7 and 11), a simple symmetric structure is the easiest to analyse. It is preferably nearly elastically isotropic. Kraynik and Warren (1994) proposed the use of a modified version of Lord Kelvin’s arrangement of tetrakaidecahedral cells, which they named as a Kelvin foam. The modified tetrakaidecahedra have planar faces, while the inter-edge angles are either 120° or 90° (Fig. 1.9a). The edges, of


13

uniform length L, link two hexagonal faces to a square face. The cells are packed in a body centred cubic (BCC) lattice. The relative density R of wet Kelvin foams can be related to the edge and face geometry when R is small. Assuming that the edge cross-sectional area A is constant, the edge relative density (or R for an open-cell foam) is Re ⫽

3A

(1.10)

2 2 L2

For closed-cell Kelvin foam, assuming a uniform face thickness δ, the face relative density Rf is given by

Rf ⫽

(3

)

2 ⫹6 6 δ 16L

(1.11)

The fraction of polymer in the faces, φ ⫽ Rf /Re. Weaire and Phelan (1994; 1996) showed that a foam, based on the dual of the metallic alloy Laves structure, has a 0.5% lower surface energy than Kelvin’s foam. This Weaire–Phelan foam (Fig. 1.2c) has eight cells in the repeat unit; its low symmetry means it has not been used to analyse foam mechanical properties. However, it has been used for air flow modelling (Chapter 8).

1.8.2 Cell (bubble) growth Models for cell growth are a precursor to some irregular cell models. Four factors affect bubble growth, whether it is in a low viscosity pre-polymer or in a high viscosity polymer melt: 1. Nucleation sites: Do bubbles nucleate at random points in the melt, or is nucleation inhibited in the immediate vicinity of growing bubbles? 2. Nucleation times: During foam extrusion (Section 3.3.1), the sudden pressure reduction as the melt passes through a die causes most nuclei to appear at approximately the same time. However, if gas is evolved by a chemical reaction, bubble nucleation may occur over a considerable period of time. 3. The kinetics of gas diffusion through the melt to the bubble affect the bubble growth law. 4. Cell coarsening: Gas diffusion between neighbouring bubbles of unequal size will cause cell coarsening, as the smaller bubble shrinks at the expense of the larger one. Alternatively, face fracture causes cells to merge.

14 Polymer Foams Handbook 1.8.3 Irregular models Two strategies are possible for generating irregular models: The first strategy is to generate a random foam structure and to check if its cell statistics are similar to those of polymer foams; Voronoi tesselations are commonly used; for example Roberts and Garboczi (2002) created three-dimensional Voronoi models of open-cell foams with a relative density of 0.05. The Poisson–Voronoi construction is shown in Figure 1.9b for a two-dimensional (2-D) tesselation of a plane (the number of nuclei in a sample volume follows a Poisson distribution). A number of random points (nuclei) are generated in a box of side L. Each cell is a convex polygon surrounding the nucleus, containing all points that are closer to the nucleus than to any other nucleus. Okabe et al. (1992) presented statistical distributions of the number of edges per cell, inter-edge angles, etc. Kumar and Kurtz (1994) fitted the distribution of the number of faces per cell in 350,000 cells with a two-parameter gamma distribution using a ⫽ 21.6 and b ⫽ 0.72 P⫽

x a⫺1 ⫺x / b e b a ⌫(a)

(1.12)

where Γ (a) is a gamma function and P is the probability of the result lying in the range x to x ⫹ dx. Alternatively they used a log-normal distribution with S ⫽ 0.21 and X ⫽ 15.2 z (001)

L

Nucleus C Equidistant line y (010) x (100) (a)

Figure 1.9

(b)

(a) Regular BCC packing of 14-sided cells in the Kelvin foam and (b) 2-D Voronoi construction for cells inside a square box.


P⫽

⎡ (ln x ⫺ ln X)2 ⎤ ⎥ exp ⎢⎢ ⎥ 2S 2 Sx 2π ⎢⎣ ⎥⎦ 1

15

(1.13)

In 3-D, random points, placed in a cubic box, act as cell nuclei. The Poisson Voronoi tesselation model assumes that: 1. all bubble nuclei appear at time zero, at random positions; 2. bubbles grow at a constant rate, independent of direction, until they impinge; 3. the bubble centres do not move during this process. However, assumptions 2 and 3 do not apply to polymer foaming processes. The volume change, as the foam expands, causes the bubble centres to move and the polymer melt in the cell edges to flow. Given the partial understanding of the four stages listed in the last section, a full model for the formation of the foam structure has not yet been developed. This subject will be revisited in Chapter 3. Kraynik et al. (2003) subjected a model dry foam, generated by a Voronoi tessellation of 512 nuclei in a cube, to series of four-stage compression and tension cycles along the three cube axes. At each stage, vertex ‘reactions’ in the deformed foam were performed to anneal the structure. A typical reaction in a 2-D foam is shown in Figure 1.10 (after Weaire, 1992), where two vertices approach, combine, and split, allowing an extension in the vertical direction. In a 3-D foam two fourfold vertices combine into an eightfold vertex before splitting again. The subsequent elimination of short edges (typically less than 10% of the mean edge length) led to foams with microstructural statistics that matched the soap froths studied by Matzke (1946). The – original Voronoi model cells had a higher average f ⫽ 14.3 and normalised surface energy E ⫽ 5.40 than the annealed foam with – f ⫽ 13.7 and E ⫽ 5.33. The distribution of cell types (characterised by the numbers of triangles, quadrilaterals, etc.) differed for the two structures. Figure 1.11 compares the initial Voronoi with the final

Figure 1.10

Vertex transformation in a dry 2-D foam.


(a)

Figure 1.11

(b)

3-D Voronoi model of (a) a closed-cell dry foam with 512 cells and (b) annealed version (b) (Kraynik et al., 2003 © The American Physical Society). 4

p (λ)

3

2

1

0

Figure 1.12

0

0.2

0.4

0.6 λ

0.8

1

1.2

Distribution of edge lengths λ in Voronoi foam (dashed line) and annealed Voronoi (solid line) (redrawn from Kraynik et al., 2003 © The American Physical Society).

annealed microstructures, while Figure 1.12 compares the monotonically decreasing edge length frequency distribution for the initial model with the near-normal distribution of the annealed structure. The latter is closer to experimental edge length distributions (Fig. 1.7a). Therefore Voronoi models, as created, are only first approximations for polymer foam structures. The annealed Voronoi models are good models for soap froths, but still differ significantly from PU foam structures (Table 1.2). The second strategy is to use the measured vertex positions of a section of real foam as the basis of a computer model. This is discussed further at the end of Chapter 7.


1.9

17

Bead foams Air channels in bead foams are the inverse of a wet open-cell foam structure, on a larger scale. The polymer structure in the latter is replaced by air channels, and the air space in the latter is replaced by polymer foam. Hence in Figure 1.2d, for a regular model, or Figure 1.6, for an irregular structure, the ‘solid’ regions should be considered as the air channels. These foams therefore have two levels of microstructure: the cell level and the bead level.

References Chan R. & Nakamura M. (1969) Mechanical properties of plastics foams, J. Cell. Plast. 5, 112–118. Elliott J.A., Windle A.H. et al. (2002) In-situ deformation of an open-cell flexible polyurethane foam characterised by 3D computed tomography, J. Mater. Sci. 37, 1547–1555. Gong L., Kyriakides S. & Jang W.Y. (2005) Compressive response of open-cell foams. Part 1. Morphology and elastic properties, Int. J. Solid. Struct. 42, 1355–1379. Ko W.L. (1965) Deformation of foamed elastomers, J. Cell. Plast. 1, 45–50. Kraynik A.D. & Warren W.E. (1994) The elastic behaviour of lowdensity cellular plastics, Chapter 7 in Low Density Cellular Plastics, Eds. Hilyard N.C. & Cunningham A., Chapman and Hall, London. Kraynik A.M., Neilsen M.K. et al. (1999) Foam micromechanics, in Foams and Emulsions, Eds. Sadoc J.F. & Rivier N., Nato ASI Series E, Vol. 354, Kluwer, Dordrecht. Kraynik A.D., Reinelt D.A. & van Swol F. (2003) Structure of monodisperse foam, Phys. Rev. E 67, 031403. Kumar S. & Kurtz S.K. (1994) Simulation of material microstructure using a 3D Voronoi tesselation, Acta Metall. Mater. 42, 3917–3927. Kusner R. & Sullivan J.M. (1966) Comparing the Weaire–Phelan equal-volume foam to Kelvin’s foam, Forma 11, 233–242. Matzke E.B. (1946) The three-dimensional shape of bubbles in foams, Am. J. Bot. 33, 58–80. Mills N.J. (2005) Plastics: Microstructure and Engineering Applications, 3rd edn., Butterworth Heinemann, London. Montminy M.D., Tannenbaum A.R. et al. (2001) New algorithms for 3-D imaging and analysis of open-celled foams, J. Cell. Plast. 37, 501–515. Montminy M.D., Tannenbaum A.R. & Macosko C.W. (2004) The 3D structure of real polymer foams, J. Coll. Interf. Sci. 280, 202–211. Okabe A., Boots B. & Sugihara K. (1992) Spatial Tesselations, Wiley, New York.

18 Polymer Foams Handbook Phelan R. & Weaire D. et al. (1996) The conductivity of a foam, J. Phys. Condens. Mat. 8, L475–L482. Plateau J. (1873) Statique Experimentale et Theoretique des Liquides Soumis Aux Seules Forces Moleculaires, Gauthier-Villars, Paris. Rhodes M.B. (1994) Characterizations of polymeric cellular structures, in Low Density Cellular Plastics, Eds. Hilyard N.C. & Cunningham A.C., Chapman & Hall, pp. 56–77. Roberts A.P. & Garboczi E.J. (2002) Elastic properties of model random 3-D open-cell solids, J. Mech. Phys. Solid. 50, 33–55. Shulmeister V. (1998) Modelling of the Mechanical Properties of Low-Density Foams, PhD thesis, Technical University of Delft, Shaker Publishing, Maastricht. Smith C.J. (1948) Grains, phases and interfaces, an interpretation of microstructure, Trans. AIME 175, 17–51. Szayna M. & Voelkel R. (1999) NMR microscopy of polyurethane foam, Solid State NMR 15, 99–102. Thomson W. (1887) On the division of space with minimum partitional area, Phil. Mag. 24, 503, reproduced in The Kelvin Problem, Ed. Weaire D., Taylor and Francis, 1996. Throne J.L. (1996) Thermoplastic Foams, Sherwood Publishers, Hickley, Ohio. Weaire D. (1992) Some lessons from soap froth for the physics of soft condensed matter, Phys. Script. T45, 29–33. Weaire D. & Hutzler S. (1999) The Physics of Foams, Clarendon Press, Oxford. Weaire D. & Phelan R.P. (1994) A counter-example to Kelvin’s conjecture on minimal surfaces, Phil. Mag. Lett. 69, 107–110. Weaire D. & Phelan R. (1996) Cellular structures in three dimensions, Phil. Trans. Roy. Soc. A 354, 1989–1997. Weaire D., Phelan R. & Verbist G. (1999) The structure and geometry of foams, in Foams and Emulsions, Eds. Sadoc J.F. & Rivier N., Nato ASI Series E, Vol. 354, Kluwer, Dordrecht. Williams R.E. (1968) Space-filling polyhedron: its relation to aggregates of soap bubbles, plant cells, and metal crystallites, Science 161, 276–277. Youssef S., Maire E. & Gaertner R. (2005) Finite element modelling of the actual structure of cellular materials determined by X-ray tomography, Acta Mater. 53, 719–730.

Chapter 2

Polyurethane foams: processing and microstructure

Chapter contents 2.1 2.2 2.3 2.4 2.5

Introduction PU chemistry PU foam processes PU microstructure Effect of microstructure on mechanical properties 2.6 PU foam microstructure Summary References

20 20 21 26 27 29 35 36


2.1

Introduction Polyurethane (PU) foams are treated separately from foamed thermoplastics (Chapter 3), because the processing differs and the foam geometry is dominated by surface tension effects. The sections consider PU chemistry, the foaming processes, the physics behind the foam structures, and then typical microstructures. Chapter 7 will consider the micromechanics of flexible PU foams, which are typically open-cell.

2.2

PU chemistry Herrington and Hock (1997) describe both slabstock and batch moulding processes for PU foams. PUs are a family of step-growth polymers made by the reaction between a di-isocyanate and a diol HO R2 OH + O C N R1 N C O O

O

O C N R1 N C O R2 H

H

n

which produces the urethane linkage NH—COO—, but no byproducts. A typical di-isocyanate is MDI (4,4⬘di-isocyanato diphenyl methane), in which the group R1 is φ CH2 φ, where φ represents a benzene ring. This forms the rigid part of the PU molecule. In a typical di-ol, R2 is a low molecular weight polyethylene oxide (M ⫽ 2000); this produces the flexible part of the PU molecule. The polyol and isocyanate components are varied to produce foams with a wide range of properties. If the di-ol is mixed with some tri-ols, based on glycerine and polypropylene oxide (PPO), the PU will be crosslinked. CO2 gas is generated by the reaction of the 1–4% water content with the di-isocyanate monomer R!N"C"O ⫹ H 2O → R!NH 2 ⫹ ↑CO2 The amide groups, formed on both ends of the monomer, react with further di-isocyanate to form urea linkages NH—CO—NH H2N CH2 CH2 NH2 + O C N R1 N C O O C

O N H

R1

N H

C

N H

CH2

CH2

N H

Chapter 2 Polyurethane foams: processing and microstructure

21

Thus the copolymer contains mostly PU with some polyurea blocks, which phase separate later in the process.

2.3

PU foam processes 2.3.1 Slabstock foam In the continuous slabstock process, it is necessary to control the reaction exotherm. If more than about 5% water is used, the foam will overheat unless special cooling technology is used (Klempner and Frisch, 2001). Therefore low-density foams may use a physical blowing agent, as well as the CO2 gas formed by the chemical reaction of water with isocyanate groups. Figure 2.1 shows the Maxfoam process in which the mixture of polyol and isocyanate is poured onto a moving conveyor belt. The foam is restricted from spreading laterally by waxed paper at the sides of the belt, and by the material behind and in front of it on the belt. Consequently, the foaming process causes the material to rise in the vertical direction – this causes the bubble shape to be anisotropic (see later). Variants of the process use gas flow through the foam, once the cell faces have burst, to remove the heat of polymerisation. PU foams are characterised by the indentation force deflection (IFD) value, which is related to the foam Young’s modulus (Section 9.3). Figure 2.2 shows that it is controlled by both the water and the methylene chloride blowing agent content, and that different densities can have the same IFD. An increase in the water content creates more polyurea ‘hard segments’, which reinforce the rubbery matrix and increases its modulus. In contrast, increasing the methylene chloride

Mixing head

Raw materials

Trough

F: Foam L: Liquid Operator’s platform

F L

Bottom paper feed

Figure 2.1

Foam Pivot points

Pivot points

Horizontal conveyor

Five-section fall plate

Maxfoam process in which the mixture of polyol and isocyanate is poured onto a moving conveyor belt (Herrington and Hock, 1997 © Dow Chemical Company).


400 pph

5

4

wat e

r

25% IFD (N)

300

3 0 3

200

5 10

100 15

Parts blowing agent

20

0 20

25 25

30

Density (kg m−3)

Figure 2.2

Slabstock foam density control by the water and the methylene chloride blowing agent content (redrawn from Herrington and Hock, 1997 © Dow Chemical Company).

content reduces the foam density, hence reducing the IFD value. The slabstock process can use polyols of relatively low average molecular mass; their lower viscosity allows cells to open at the end of the reaction, if suitable surfactants are used (Weier and Burkhart, 2001). Neff and Macosko (1995) describe four stages in the shear modulus vs. time graph of the flexible PU foam process. In the first stage, bubbles are generated in a low viscosity liquid, and the dynamic shear modulus rises rapidly in the first 20 s. In the second stage, it reaches a plateau level of about 20 Pa (Fig. 2.3), the modulus of the fully expanded wet foam. Reinelt and Kraynik (1996) calculated the average shear modulus G of soap foams, using a uniform-sized Kelvin cell model, as G ⫽ 0.81

T V1/ 3

(2.1)

where T is the surface tension of the fluid and V is the cell volume. Using T ⫽ 40 mN/m and cell diameter V1/3 艑 0.5 mm gives G 艑 60 Pa, the correct order of magnitude. The initial low viscosity of the reactants


23

0.2 1

104

2

3

4 G′

G′, G′′ (Pa)

1000

G′′

0.1

100

Bi-dentate urea (FTIR, ν = 1640 cm−1) 0.05

10

1

0.15

0

50

100

150

Relative IR peak area

Cells open

0 200

Time (s)

Figure 2.3

Four stages in the development of the shear modulus of a flexible PU foam (Neff and Macosko, 1995 © Society of the Plastics Industry).

means that surface tension forces determine the foam shape (Chapter 1). Fourier transform infra red (FTIR) spectroscopy showed that the shear modulus rise, in the third stage of Figure 2.3 starting at 70 s, was due to phase separation into a rubbery matrix and a rigid urea phase. The modulus increase in the fourth stage indicates the crosslinking of the polymer matrix. The final phenomenon in most PU foam processing is cell opening; this instability has been varyingly attributed to surfactants, phase precipitation, and excessive biaxial stretching. Yasunaga et al. (1996), using a parallel-plate rheometer, linked the sudden drop in the normal force on the plate to the onset of cell face fracture. Although some fragments of broken faces were observed, the material in other faces flowed back into the edges. Thus, the face material must be semi-solid at the time of cell face fracture. Silicone surfactants in the PU formulation control the amount of open faces and the average cell size. Scanning electron microscope (SEM) shows that some faces have incipient holes around their periphery (Fig. 2.4a). The air flow through the foam correlates with the fraction of open-cell faces seen by SEM. Hence, air-flow measurements (Chapter 8) are used to characterise PU foams. Figure 2.4b shows a range of cell face types. To make reticulated foams, a mixture of hydrogen and air is allowed to diffuse into the cells, then exploded to remove the faces, leaving just the cell edges. Reticulated foams for filter applications usually have large cells to allow easy air flow; such foams can be coated to make ceramic or metal foams, for use as high temperature filters and in heat exchangers. Consequently, the metal versions, such


(a)

Figure 2.4

(b)

(a) Cell face fracture starts from thin regions at the edges of faces (Weier and Burkhardt, 2001 © Rapra) and (b) a variety of cell face types arrowed – open, partly open and closed (Yasunaga et al., 1996 © Sage).

as ‘Duocell’, have cell shapes and foam-relative densities similar to those of reticulated PU foams.

2.3.2 Moulded PU foam High resilience (HR) foam has been used for the majority of (US) automotive seat cushions since the 1980s. The polyether tri-ols have molecular weight range of 4000–6000 Da with a 5–25% end capping with ethylene oxide. The primary hydroxyl content, which can vary from 65% to 90%, affects the reaction rate. There is less need for a gelation catalyst than with slabstock foam, because both the initial molecular weight and the functionality of the tri-ol are higher. The isocyanate can be all TDI, or a blend of TDI with a polymeric form of MDI. Silicone surfactants are used to control the cell size, edge thickness, and amount of cell face rupture. Because polymerisation occurs in a closed mould, at a faster rate than in the slabstock process, demoulded seats contain a significant proportion of closed cells. They are mechanically crushed soon after demoulding to fracture cell faces and prevent dimensional shrinkage. In this cyclic process the control of the cycle time is important for productivity. Dounis and Wilkes (1997) describe the chemical differences between slabstock and one-shot foam mouldings. In the moulded foams, the gel fraction is slightly lower, and there is less short-range ordering in the hard segment domains. The amorphous content was greater in a system with the same water and TDI content.

Chapter 2 Polyurethane foams: processing and microstructure −30

−20

−10

0

10

20

25

30 1.2

9.0 1.0 0.8 8.0

Densified PU foam CF47 Frequency 1 Hz Heating rate 1°C min−1

0.6

7.5

Tan δ

Log modulus (Pa)

8.5

0.4

7.0

0.2

6.5

0.0 −30

−20

−10

0

10

20

30

Temperature (°C)

Figure 2.5

Storage Young’s modulus and tan δ vs. temperature for densified Confor 47 PU foam (Davies and Mills, 1999).

The ethylene oxide capping of the polyols is believed to promote compatibility between the hard and soft segments. The hard regions are only held together by weaker mono-dentate hydrogen bonds, whereas in slabstock foams there is stronger bi-dentate hydrogen bonds with the urea regions. Consequently, at high temperatures and humidity, the moulded foams exhibit more compression set and other viscoelastic phenomena. The use of diethanol amine as a release agent in moulded foams is believed to prevent the full development of local packing in the hard segments.

2.3.3 Slow-recovery foams Slow-recovery PU foams, used for example in wheelchair seating, are usually made by a batch process. Hence they are a variant of moulded PU foams. Davies and Mills (1999) carried out flexural tests on densified Confor 47 foam, of density 1000 kg m⫺3. The storage Young’s modulus of 1 GPa at ⫺30°C (Fig. 2.5) is typical of a polymer in a glassy state, whereas that of about 5 MPa at 50°C is typical of a rubber. The maximum value of tan δ occurs at 20°C, the glass transition temperature Tg. The height (1.1) of the tan δ peak is greater than that for the flexible PU foam (0.4) measured by Turner et al. (1984).


Smeared absolute intensity

0.015

No copolymer polyol; no diethanol amine No copolymer polyol; with diethanol amine

0.010

0.005

0.000 0.000

0.005

0.010

0.015 s

Figure 2.6

0.020

0.025

0.030

[Å−1]

Small angle X-ray scattering of moulded flexible PU foams with and without diethanol amine crosslinking agent (Kaushiva and Wilkes, 2000, © Elsevier).

Section 2.5 describes how the hard blocks in PUs act as a microcrystalline phase, stiffening the rubbery matrix.

2.4

PU microstructure Neff and Macosko (1995; 1996) and Zhang et al. (1999) investigated the microstructure development of PU foams, building on knowledge of the microstructure of solid PU (Armistead et al., 1988). Transmission electron microscopy shows urea-rich regions roughly 100 nm diameter which act as hard inclusions. Wide angle X-ray diffraction patterns show an amorphous halo for a spacing of 0.4–0.6 nm, and sometimes a diffuse halo for a 1.6 nm spacing. Small angle X-ray scattering patterns indicate a periodicity in the structure on a 5–10 nm scale (Fig. 2.6) in moulded PU foams (that do not contain diethanol amine crosslinking agent). A number of techniques have been used to detect the hard segments. Atomic force microscope (AFM) images (Fig. 2.7) show there is a larger scale inhomogeneity in the moulded foam that does not contain diethanol amine. The combined evidence does not indicate conventional crystallinity (as in low density polyethylene (LDPE) or polyethylene polypropylene (PP)), rather than there is a fine structure of the type shown in Figure 2.8. Structure development in the solid PU is related to phase separation in block copolymer systems by spinodal decomposition (Mills, 2005).

27


40.0°

25.0°

20.0°

12.5°

0.0°

0.0°

50 nm

50 nm

(a)

(b)

Figure 2.7

AFM images moulded flexible PU foam (a) without and (b) with diethanol amine cross linking agent (Kaushiva and Wilkes, 2000, © Elsevier).

Continuous soft phase

Copolymer polyol (0.5 µm)

Hard-segment (∼50 Å)

Covalent network structure

Figure 2.8

2.5

Schematic microstructure of hard and soft regions in PUs (Herrington and Hock, 1997 © Dow Chemical Company).

Effect of microstructure on mechanical properties The shear modulus G of a crosslinked rubber is related to the density of network chains per unit volume N through the classical relationship of rubber elasticity theory G ⫽ NkT

(2.2)


Stress (nominal) (MPa)

45 40 35 30 25 20 15 10 5 0 1

Figure 2.9

1.5

2

2.5 Elongation (λ)

3

3.5

4

Tensile response of a single edge, with hard domain content 31.4%, compared with theoretical curve for rubber elasticity (Van der Heide et al., 1999 © Wiley).

where k is Boltzmann’s constant and T is the absolute temperature. The level of crosslinking, hence N, can be determined from the gel content of the PU; however, the modulus predicted using equation (2.2) is two or three orders of magnitude too low. The higher modulus is due to rigid, urea phase, inclusions reinforcing the PU rubber (Fig. 2.8). In polyethylene, where crystals have a similar effect, the logarithm of Young’s modulus is proportional to the volume fraction crystallinity. The only publication found on the mechanical response of a single PU foam cell edge was by Van der Heide et al. (1999). Figure 2.9 shows the tensile response, measured at a low strain rate. The theoretical curve for extension ratios ⬎1.2 is based on the rubber elasticity theory. The response can be interpreted as corresponding to an initial yield stress of 10 MPa. Although there have been no cyclic loading tests on single foam edges, the response is expected to be similar to that of thermoplastic PUs, which have related microstructures with hard and soft segments. Figure 2.10 shows a number of compressive cycles at a slow strain rate. The response changes after the first cycle and settles down to a cycle in which there is still considerable hysteresis. The repeated high strain loading of a foam edge is expected to show similar phenomena. Laity et al. (2006) showed that the hard segments in thermoplastic PUs partly align with the tensile direction as the polymer is stretched. However, the dramatic change in the response between the first and second cycle may be due to slippage of entanglements in the first cycle. Compared with LDPE (Chapter 3), flexible PUs have a lower initial yield stress. However, in many ways they act like a semicrystalline polymer with a crystallinity of circa 20%, that is above its glass transition temperature.


29

20 N=1 N=2 N=4

True stress (MPa)

16

N=4

12 N=2 N=1 8

4 N = 1, 2, 4 0

0

0.25

0.5

0.75

1

True strain

Figure 2.10

Cyclic compression of a thermoplastic PU, with the cycle number N marked (Qi and Boyce, 2005 © Elsevier).

van der Schuur et al. (2004) studied the mechanical properties of slabstock polyether (urethane–urea) foams that had been compression moulded into a fully dense solid. The urea content was varied by varying the toluene di-isocyanate (TDI) to water ratio in the feed. Dynamic mechanical thermal analysis (DMTA) (Fig. 2.11) of a polyether urethane based on PPO shows how the shear modulus increases with the hard segment content in the temperature region above the Tg of the rubbery phase (⫺50°C) and the below melting temperature of the hard segments (circa 250°C). This modulus increase is similar to that caused by crystals in the rubbery matrix of LDPE at room temperature. By changing the polyol and isocyanate, the glass transition of the PU can be altered.

2.6

PU foam microstructure The various categories of flexible, low-density PU foams differ in microstructure. The whole spectrum from open- to closed-cell structures is possible; holes in the middle of faces can vary in size, or there can be a combination of open and closed faces.

30 Polymer Foams Handbook 10,000

100,000

10,000

1000

25.8

100

30.4 34.6

10

1

G ′′ (MPa)

G ′ (MPa)

1000 100

10

25.8

1

30.4 0.1 −100

Figure 2.11

34.6 0

100 Temperature (°C)

0.1 200

300

Dynamic storage and loss shear moduli vs. temperature, for compacted polyester PU with PPO, at the % urea content labelled (van der Schuur et al., 2004 © Elsevier).

2.6.1 Slabstock PU foams The microstructure of PU foams was described by Armistead et al. (1988). Image analysis of cell size distributions was published by Schwartz and Bomberg (1991), Lewis et al. (1996), and Hamza et al. (1997), while X-ray CT determinations of cell diameter and edge length distributions were described in Chapter 1. Gong et al. (2005) measured the edge width variation in a PU foam (Fig. 1.3), while Figure 2.12 shows the face thickness distribution measured from interference patterns of monochromatic light. In the wet Kelvin foam model, the edge breadth at mid-length bmin can be related to the edge length L and the foam relative density R. Equation (1.10) gives a similar relationship if the edges have a constant cross-section. However PU foams have a range of cell sizes, and the relationship between R and the mean values of bmin and L is not known. Furthermore, if only the longer edges in the PU structure are measured, the average b/L ratio will be overestimated.

2.6.2 Moulded foams Wolfe (1982) attributed high resilience to thick cell edges, and a proportion of closed cells in PU foams. The microstructure of HR foams (Kleiner et al., 1984) contains a range of cell sizes (Fig. 2.13). Hager et al.


31

0.834 µm 1.00 µm 1.334 µm 1.50 µm 1.667 µm 1.834 µm 2.00 µm

(a)

Grab

(b)

Figure 2.12

(a) Interference pattern from a flexible PU foam cell face with light of wave length 700 nm and (b) thickness contours of the face (Yasunaga et al., 1996 © Sage).

Figure 2.13

PU foam with a range of cell sizes: white scale bar is 1 mm (Kleiner et al., 1984 © Sage).

(1990) attributed the more-linear IFD vs. deflection graphs of Union Carbide Ultracell foams to a range of cell sizes, hence a range of cell strengths, so the foam structure collapses progressively. In contrast the cells in conventional foams have a narrow cell and diameter


(a)

Figure 2.14

(b)

Crushed moulded foam (Herrington and Hock, 1997) © Dow Chemical Company.

distribution, consequently all the cells collapse in compression within a narrow load range (see Chapter 5). Dounis and Wilkes (1995) describe SEM of moulded foams, which had less anisotropic cell shapes than slabstock foams. The cell edges were in general thicker, and more cell faces were intact, which decreased the air-flow rate through the foam by a factor of 5 compared with a slabstock foam. Herrington and Hock (1997) show SEM photos of moulded foams after they have been crushed. This fractures the fragile cell faces, but also introduces cracks (Fig. 2.14), making the foams less resistant to fatigue.

2.6.3 Rebonded PU foams The slabstock process for PU flexible open-cell foam produces some scrap, which is diced into pieces of 5–10 mm diameter. Pieces of different colours and hardnesses are randomly mixed, with more of the isocyanate and polyol chemicals added, and remoulded under uniaxial compression. British Vita (Kay Metzeler, Ashton-under-Lyne) is a major UK source of such remoulded (or ‘chip’) PU foam. Figure 2.15 shows the edge of a compressed chip in a judo mat of density 100 kg m⫺3 in which the cell edges are significantly compressed. The microstructure, with pre-bent cell edges, provides a more-linear compressive stress–strain response than virgin PU foam. There is also some closed cell material, part of the extra PU used to bond the chips together. The variation in the mechanical properties of 100 mm cube samples with position in the block was ⫾5%, since each sample contains a large number of chips. The elastic anisotropy was found to be the order of 10%.


Bonding material

Figure 2.15

33

Compressed chip

SEM of unloaded chip foam of density 113 kg m⫺3 from a judo mat (Mills, 2003). Scale bar is 1 mm.

2.6.4 Slow-recovery PU foams Slow-recovery foams have relatively high densities, with near-spherical cell shapes, and circular holes between the faces (Fitzgerald et al., 2004). In Table 2.1 the relative density values are based on a nominal PU solid density of 1200 kg m⫺3. The Royal Medica (Italian) foam formed the lower part of a ‘Dynamic’ wheelchair cushion. The Sunmate slowrecovery foams, from Dynamic Systems Inc (Leicester, NC, USA), respond to a heat stimulus by softening; this is claimed to relieve localised high sitting pressures. The much denser Pudgee foam has a different microstructure in which the bubbles maintain their spherical shape. The features that affect air flow are the size and number of holes in cell faces, their orientation and spacing, and the cell size. SEM allows the observation of the exterior of a sectioned foam; the sectioning plane cuts through the structure, leaving partial cells, faces, and holes in faces. Figure 2.16 shows features that should be considered: (a) Cut cell boundaries consisting of linked faces and vertices. Where the section passes through a hole in face, the hole is filled in to


(a)

(b)

(c)

(d)

Figure 2.16

SEM of PU foams: (a) Royal Medica, (b) Pudgee, (c) Sunmate soft, and (d) Sunmate medium.The sheet normal direction is vertical (Fitzgerald et al., 2004)

Table 2.1 Cell parameters for some slow-recovery PU foams

Foam Royal Medica Sunmate soft Sunmate medium Sunmate firm Pudgee

Sectioned cell – diameter DS 102 µm

Hole diameter – DH 102 µm

Solid area AV %

– – DH/DS

Relative density R

3.0 ⫾ 1.1 2.9 ⫾ 1.8 2.8 ⫾ 1.0

0.78 ⫾ 0.52 0.51 ⫾ 0.041 0.34 ⫾ 0.27

3.3 ⫾ 1.4 1.7 ⫾ 0.7 1.7 ⫾ 0.6

0.26 0.18 0.12

0.108 0.072 0.066

2.9 ⫾ 0.9 3.3 ⫾ 1.8

0.29 ⫾ 0.15 0.87 ⫾ 0.75

1.1 ⫾ 0.4 9.2 ⫾ 1.3

0.10 0.26

0.069 0.185


35

complete the cell perimeter. The diameter of the equivalent circle (the circle with the same included area) was computed for each cell (or hole). (b) Face holes. No allowance is made for face tilt, relative to the sectioning plane, in estimating distances or areas. (c) The area fraction of solid polymer (mainly in vertices) AV in the sampled cross-section. It is not possible to directly estimate the relative volume of polymer in the cell faces RF, since individual faces vary in thickness, and high magnification images are required to measure the face thickness. Fitzgerald et al.’s (2004) results (Table 2.1) relate to SEM images similar to those in Figure 2.16. The mean diameter of sectioned cells is close to 300 µm for all the foams. However, most sections of a cell are smaller than its maximum cross-section. If the cells were uniform – in size, the mean cell diameter DC would be 1.62 times the mean – diameter of sectioned cells DS, assuming that sections occur at random positions. Therefore, the mean cell diameter is approximately 0.5 mm for all the foams. The volume fraction of polymer in vertices is approximately equal to the area fraction AV measured on sections. Hence, for the Sunmate foams with R ⫽ 0.07, the face relative volumes RF are 0.05–0.06. RF for the Pudgee foam is 0.09 while that for the Royal Medica foam is 0.08. These values are comparable, so the foam density differences are mainly due to variation in the vertex content. The ratio of the mean hole diameter to the mean sectioned cell – – diameter DH /DS (Table 2.1) is smaller for the Sunmate foams than for – – the others. These values can be corrected to get DH /DC, since the cell – – diameter DC is approximately 1.62 DS.

Summary The main process influences on PU foams are: 1. Surface tension determines the cell shape. The cells can be elongated if the foam rises, as in the slabstock process (Fig. 2.1). 2. The chemicals used determine the microstructure of the PU. 3. Silicone surfactants control cell face opening; it can vary from zero, through partial, to 100%. The geometry of foam cells is quantified to assist the modelling of mechanical properties. Statistical descriptions are necessary because of the variability in the structure.


References Armistead J.P., Wilkes G.L. & Turner R.B. (1988) Morphology of water-blown flexible polyurethane foams, J. Appl. Polym. Sci. 35, 601–629. Davies O.L. & Mills N.J. (1999) The rate dependence of Confor PU foams, Cell. Polym. 18, 117–136. Dounis D.V. & Wilkes G.L. (1995) A structure property comparison between flexible molded polyurethane foams and conventional slabstock foams, Polyurethanes Conference, 353–361. Dounis D.V. & Wilkes G.L. (1997) Structure property relationships of flexible polyurethane foams, Polymer 38, 2819–2828. Fitzgerald C., Lyn I. & Mills N.J. (2004) Air flow through polyurethane foams with near-circular cell face holes, J. Cell. Plast. 40, 89–110. Gong L., Kyriakides S. & Jang W.Y. (2005) Compressive response of open-cell foams, Part 1. Morphology and elastic properties, Int. J. Solid. Struct. 42, 1355–1379. Hager S.L., Kilgour J.A. & Schiffauer R. (1990) ULTRACEL foam technology: high performance slabstock foam without CFCs or methylene chloride, J. Cell. Plast. 26, 258–273. Hamza R., Zhang X.D. et al. (1997) Imaging open-cell polyurethane foam through confocal microscopy, ACS Conference, 165–177. Herrington R. & Hock K., Eds. (1997) Flexible Polyurethane Foams, 2nd edn., Dow Chemical Company, Midland, MI. Kaushiva B.D. & Wilkes G.L. (2000) Alteration of polyurea hard domain morphology by diethanol amine in molded polyurethane foams, Polymer 41, 6981–6986. Kleiner G.A., Pham T. & Tenhagen R.J. (1984) Recent advances in high resilience foam technologies, J. Cell. Plast. 20, 49–57. Klempner D. & Frisch K.C. (2001) Advances in Polyurethane Science, Rapra, Shawbury. Laity P.R., Taylor J.E. et al. (2006) Morphological behaviour of thermoplastic polyurethanes during repeated deformation, Macromol. Mater. Eng. 291, 301–324. Lewis K.M., Kijak I. et al. (1996) An image analysis method for cell-size and cell-size distribution in rigid foams, J. Cell. Plast. 32, 235–259. Mills N.J. (2003) Foams in sport, Chapter 2 in Sport Materials, Ed. Jenkins M.J., Woodhead, Cambridge, pp. 9–46. Mills N.J. (2005) Plastics: Microstructure and Engineering Applications, 3rd edn., Butterworth Heinemann, London. Neff R. & Macosko C.W. (1995) A model for modulus development in flexible polyurethane foam, Proceedings of the SPI Polyurethanes Annual Tech Marketing Conference, pp. 344–352. Neff R. & Macosko C.W. (1996) Simultaneous measurement of viscoelastic changes and cell opening during processing of flexible PU foam, Rheol. Acta 35, 656–666.


37

Qi H.J. & Boyce M.C. (2005) Stress–strain behaviour of thermoplastic polyurethanes, Mech. Matl. 37, 817–839. Reinelt D.A. & Kraynik A.M. (1996) Linear elastic behavior of dry soap foams, J. Coll. Interf. Sci. 181, 511–520. Schwartz N.V. & Bomberg M. (1991) Image analysis and the characterisation of cellular plastics, J. Therm. Insul. 15, 153–171. Turner R.B., Spell H.L. & Wilkes G.L. (1984) Dynamic mechanical spectroscopy study of flexible urethane foam, Proceedings of the 28th SPI Annual Tech/Marketing Conference. Van der Heide E., van Asselen O.L.J. et al. (1999) Tensile deformation behaviour of the polymer phase of flexible polyurethane foams and polyurethane elastomers, Macromol. Symp. 147, 127–137. van der Schuur M., Van der Heide E. et al. (2004) Elastic behaviour of flexible polyether(urethane–urea) foam materials, Polymer 45, 2721–2727. Weier A. & Burkhart G. (2001) Demands on surfactants in polyurethane foam production, Chapter 2, in Advances in Urethane Science and Technology, Eds. Klemperer D. & Frisch K., RAPRA, Shawbury. Wolfe H.W. (1982) Cushioning and fatigue, in Mechanics of Cellular Plastics, Ed. Hilyard N.C., MacMillan, London. Yasunaga K., Neff R.A. et al. (1996) Study of cell opening in flexible polyurethane foam, J. Cell. Plast. 32, 427–448. Zhang J., Macosko C.W. et al. (1999) Role of silicone surfactant in flexible PU foam, Coll. Interf. Sci. 216, 270.

Chapter 3

Foamed thermoplastics: microstructure and processing

Chapter contents 3.1 Introduction 3.2 Polyolefins 3.3 Processing 3.4 Foam crystallinity and crystal orientation Summary References

40 41 46 62 64 64


3.1

Introduction This chapter describes polymer foam microstructures and explains how they are related to foaming processes. Descriptions of the foam geometry and the polymer microstructure, if quantitative, can be used in modelling to predict relationships between microstructure and mechanical properties. The foam behaviour depends on the properties of the polymer (Table 3.1) from which it is made. Nearly all thermoplastics can be foamed, but commercially important foams are mainly based on commodity plastics – PE, PP, PS, and PVC. This partly reflects their low cost, and partly the availability of grades with highly elastic melts. Many textbooks, for example Mills (2005), explain the structure–property relationships for polymers. Polyolefins were selected to illustrate thermoplastic foam matrices; a recent review (Mills, 2004) gives an extensive bibliography. Olefins are unsaturated aliphatic hydrocarbons. For foams, ethylene and propylene are the main monomers; their polymers, copolymers, and blends with other polymers are used to produce polyolefin foams with a range of crystallinity. Certain melt rheological properties are necessary for successful melt inflation into thin-walled foam cells; molecular structures are tailored to achieve these properties. The processing section shows how, with suitable additives and process control, stable foams can be produced. Polymer chemists and technologists have played important roles in these developments. Benning (1967) reviewed the development of closed-cell polyolefin foams, and their mechanical properties. Some of his predictions on materials development turned out to be true. In Part I, he explained that non-crosslinked PE foam melts have inferior tensile creep properties to crosslinked foams. He predicted that a range of new foams could be made using random or graft PE copolymers. Extruded foams could be made with fine cells, in sheet and tube form, and as insulation on wire. He showed the effects of gel content on the foam density. Part II describes the shape of the compressive stress–strain graph, with the

Table 3.1

Thermoplastics commonly used for closed-cell foams Polymer name

Abbreviation

Type

Polyethylene Ethylene–vinyl acetate copolymer Polypropylene Polystyrene Polyvinylchloride Polyphenyleneoxide blend with PS

PE EVA PP PS PVC PPO

Semi-crystalline Semi-crystalline Semi-crystalline Glass Glass⫹10% crystalline Glass

Chapter 3 Foamed thermoplastics: microstructure and processing

41

initial elastic region, plateau region, and an upturn at high strains. Graphs showed how the foam flexural modulus depends on a power of its density, a result taken up 20 years later by Gibson and Ashby (1988). Part III emphasised cell face orientation, noting that faces shrivel on a hot stage microscope. Much subsequent research has confirmed his predictions and provided details of the technology and science. The chemical resistance of polyolefin foams is good, given the good resistance of the solid polymers to acids, alkalis, and solvents.

3.2

Polyolefins 3.2.1 PEs and copolymers In the 1990s PEs could be divided into high-density and low-density polyethylenes (LDPE and HDPE, respectively), made respectively by low-pressure (circa 20 bar) Ziegler–Natta process, and high-pressure (circa 200 bar) ICI process. Eaves (2001) distinguished between polyolefin plastomers (POP) with densities ⬎910 kg m⫺3 and polyolefin elastomers (POE) with densities ⬍910 kg m⫺3. The density of a PE at 20°C is a linear function of the crystallinity, with limiting values of 854 kg m⫺3 for zero crystallinity and 1000 kg m⫺3 for 100% crystallinity. LDPE contains long chain branches, which give the melt a high elasticity, hence a high tensile strength. Linear low-density polyethylene (LLDPE) can be produced by using Ziegler–Natta catalyst systems at low pressures; it has a narrower molecular weight distribution (MWD) than LDPE. The more recent metallocene single site catalysts (Kristen, 1999) allow the production of ethylene copolymers with larger amounts of octene than with Ziegler catalysts, hence densities lower than LDPE (910 kg m⫺3) can be made. Figure 3.1 shows how the % crystallinity varies with the comonomer concentration. The Young’s modulus of PE in spherulitic form increases with crystallinity. The modelling of spherulite mechanics is complex; in some regions the lamellar crystals and amorphous interlayers act in parallel, and in others they act in series. Consequently no complete models exist, but the logarithm of the Young’s modulus increases almost linearly with the % crystallinity (Fig. 3.2). As the Young’s moduli of the two phases (amorphous and crystalline) are so different, there is a marked increase in modulus with crystallinity. The initial tensile yield stress of spherulitic PE also increases markedly with crystallinity, as the yield strain remains almost constant in the range 5–10%. Biaxially drawn polymer films have a similar microstructural state to polyolefin foam faces (Mills and Zhu, 1999). If the polymer melt deforms entirely elastically in the foaming process, the biaxial extension ratio is 4.6 for a relative density of 0.05, and 6.3 for a relative density


% Crystallinity

80 Ziegler Metallocene

60

40

20

0

0

20

40

60

80

Branches per 1000 C

Figure 3.1

Crystallinity of PE copolymers vs. the number of short branches per 1000 C atoms. (C4 comonomers for metallocene, C4 and C6 comonomers for Ziegler. Data from Mirabella and Bafna (2002) J. Polym. Sci. B 40, 1637.) 40 30 20

Young’s modulus (GPa)

1

0.1

0 0.3

0.4

0.5

0.6

0.7

0.8

Crystallinity

Figure 3.2

Predicted Young’s modulus of spherulitic PE vs. crystallinity, for lamella of various aspect ratios, compared with experimental data from Crist et al. (1989) Macromolecules 22, 1709. (Guan and Pitchumani (2004) Poly. Eng. Sci. 44, 433 © Wiley.)

of 0.02. Figure 3.3 shows the tensile stress–strain curve of LDPE packaging film, of density 910 kg m⫺3 and thickness 45 µm, having a biaxial draw ratio of 6.5. The film yields at a stress of 10 MPa at a strain of 8%, and the stress rises linearly to 20 MPa at a strain of 90%.


80

0

5

10

15

43

20 25

b 20

Stress (MPa)

60

15 40

a 10

20 5 0

0

20

40

60

80

0 100

Strain (%)

Figure 3.3

Stress–strain curve for the tensile deformation of oriented polymer films for (a) 45 µm thick LDPE with an extension ratio of 6.5 and (b) 32 µm thick PS with a biaxial extension ratio of 5.5 (Mills and Zhu, 1999).

Its Young’s modulus up to 1.5% strain, measured at a strain rate of 7 ⫻ 10⫺4 s⫺1, was 202 MPa. The sample extends uniformly, unlike LDPE film of lower orientation, which necks and cold draws. No necks were observed in the faces of deformed LDPE foams, so their yielding response is similar. Figure 3.3 shows the tensile stress–strain curve of Dow Window Film 6003E of thickness 32 µm, measured in the machine direction. This biaxially oriented PS film, of draw ratio 5.5, yields at a stress of 70 MPa and strain of 3.5%, and then extends nearly uniformly at a constant yield stress. In contrast, bulk unoriented PS fails by crazing at a stress of about 40 MPa. The film Young’s modulus, up to 1% strain, is 3.08 GPa at a strain rate of 7 ⫻ 10⫺4 s⫺1, the same as for unoriented bulk PS. Conventional LDPE has a wide MWD. LDPE made using metallocene catalysts has a narrow MWD (MW/MN ⫽ 2), so may have a much lower level of melt elasticity. Metallocene chemistry also allows the production of copolymers with a larger comonomer content in the high molecular weight part than in the low molecular weight part; this can be achieved in a single reactor by combining two metallocene catalysts. Metallocene PEs crystallise differently than Ziegler–Natta PE; due to the more uniform microstructure and narrower MWD, the crystalline lamellae are uniformly thin and have slightly lower melting points than Ziegler PE. The different crystalline morphology affects the mechanical properties. In the USA, three firms (Dow, Exxon Mobil, and Nova Chemical) use the metallocene catalysis route for production of PE.

44 Polymer Foams Handbook POE foams compete with ethylene–vinyl acetate copolymer (EVA) foams. The VA content is usually between 15% and 22%, which means that the crystallinity is between 25% and 10%, lower than the 40–50% for LDPEs. The crystalline regions in EVA have a melting temperature of about 70°C, compared with about 110°C in metallocene LDPE. Figure 5.3 shows the dynamic modulus and damping as a function of temperature; the Tg is ⫺10ºC for the 18% VA copolymer shown. Bistac et al. (1999) measured the damping peaks of EVA for a range of composition.

3.2.2 Blends Rodriguez-Perez et al. (1998) investigated the effect of blending LDPE with EVA or a styrene–isoprene block copolymer. The thermal expansion coefficient, Young’s modulus, and thermal conductivity of the foamed blends usually lay between the limits of the foamed constituents, although the relationship between property and blend content was not always linear. The reasons must lie in the microstructure. Most polymer pairs are immiscible, and for these the majority phase tends to be continuous. The morphology of blends of EVA and metallocene catalysed ethylene–octene copolymer depends on the EVA content (Kontopoulou et al., 2003). With 25% EVA, the EVA phase occurs as fine spherical inclusions in the LDPE matrix.

3.2.3 Ethylene styrene ‘interpolymers’ Ethylene styrene ‘interpolymers’ (ESI) were produced by Dow using metallocene catalysts. The term ‘interpolymer’ implies a random copolymer. Some can be amorphous and glassy at room temperature, so differ markedly from other polyolefins, which are semi-crystalline and above Tg at room temperature. They are relatively expensive; consequently these materials were blended with cheaper LDPE or EVA. Dow originally produced these materials, then a joint venture Select produced ESI foams. Nova chemicals Arcel is a 70% styrene 30% ethylene interpolymer used for bead foam moulding. The styrene content affects the crystallinity of ESI (Chaudhary et al., 2000); for ⬎50% styrene the copolymers are amorphous. As the styrene content is increased from 50% to 70%, the glass transition temperature Tg increases from ⫺15°C to 20°C. Low-density foams were made (Liu and Tsiang, 2003) from a blend of 50% of various ESI polymers, 33% of EVA, and 17% of azodicarbonamide blowing agent. Thermal analysis showed that the blends, with ESI containing 70% styrene, had a Tg in the range 22–30°C. Dynamic mechanical thermal analysis (DMTA) traces show that these blends soften over


45

the temperature range 20–40°C, so have high damping in this range. The relative densities of these foams were approximately 0.04; at 20°C they have tensile yield stresses in the range 1.5–2.5 MPa. Semi-crystalline LDPE is immiscible with amorphous ESI. Consequently, in LDPE/ESI blends, there are rigid crystalline PE and rubbery amorphous PE regions, mixed on a 0.1 µm scale, together with regions of leathery ESI on a 5–10 µm scale (Ramesh, 2000). As the 70% styrene ESI was amorphous, increasing its content reduced the overall crystallinity of the blend. However, with 25% ESI, the compressive stress at 25% compression of the foam is unchanged. The processing window with ESI is wider, and the cell structure is slightly finer, than for the equivalent LDPE foam. Ankrah et al. (2002) investigated foamed blends of LDPE with up to 40% of ESI (70% styrene). These had slightly lower initial compressive yield strengths than the LDPE alone, allowing for the density of the foam, with a similar temperature dependence of the yield stress (Fig. 3.4). Although the yield stress is higher than EVA foam of the same density, the compression set values are lower. The ESI/LDPE foams have improved impact properties, compared with EVA foams of similar density. Analysis of creep tests shows that air diffuses from the cells at a similar rate to EVA foams of a greater density.

300 LD45 20% ESI in LDPE 30% ESI in LDPE 40% ESI in LDPE EVA 30

Initial yield stress (kPa)

250

200

150

100

50

−10

0

10

20

30

40

Temperature (°C)

Figure 3.4

Variation of initial compressive yield stress of ESI/LDPE foams with temperature, compared with EVA and LDPE foams; all corrected to a density of 47 kg m⫺3 (Ankrah et al., 2002).

46 Polymer Foams Handbook 3.2.4 Ethylene–propylene–diene monomer The majority of these systems are crosslinked, so are thermosets; the term thermoplastic vulcanisates (TPV) is sometimes used. Copolymers are made from ethylene, propylene and a small fraction of unsaturated diene, allowing the crosslinking of the foam. For applications such as mouse mats, foam of a moderate density is loaded with carbon black to prevent static build-up. The optimisation of ethylene–propylene–diene monomer (EPDM) processing is complex (Krusche and Haberstroh, 1999), since the blowing agent decomposition and the crosslinking reactions may influence each other. High-activity zinc oxide is used to accelerate the crosslinking reaction, necessary for the production of weatherstrips.

3.2.5 Polypropylenes It is difficult to foam linear PP, since it typically has a lower viscosity and lower melt elasticity than LDPE, causing cell faces to fracture. Special branched PP grades, with high melt elasticity, were developed for foam extrusion (Phillips et al., 1992). The more common technology for making high melt strength PP uses post-polymerisation treatments to graft branches to the main chain. Alternatively, branched PP can be made directly in the polymerisation reactor. Dow ‘Inspire’ PP (Thoen et al., 2002) is based on the ‘Insite’ polymerisation technology, but no details are given either of the catalysts or the resulting molecular architecture. Section 3.3.2 explains the characterisation of melt elasticity. The low melt elasticity of linear PP results from the lack of long chain branching and the relatively narrow MWD (the melt is less stable than PE, and melt degradation can narrow the MWD to a MW/MN ratio of 2). Higher melt temperatures are required for processing PP foam, than for PE foam, since the crystalline melting temperature is 170°C. Melt degradation is likely for PP foam, since the foam has a high surface area and PP is less thermally stable than PE. Consequently the processing temperature window is small. The effects of butane level and melt temperature on PP foam density (which can be as low as 15 kg m⫺3) were explored by Folland et al. (2002). The homopolymer foam is rather stiff and brittle, but blends made with a PP block copolymer have reduced modulus and increased toughness; however there is a maximum copolymer content for the production of low-density foam.

3.3

Processing Park (2004) and Gendron (2005) cover the processes for polyolefin foams and the blowing agents used. The compression moulding of


47

EVA foam shoe midsoles use a different technology, with crosslinking and expansion in a heated mould. Eaves and Witten (1998) described the Zotefoams process, in which nitrogen is dissolved into molten crosslinked polyolefin sheets in a high-pressure autoclave, which are then expanded into foams in two stages. Recent research has concentrated on process refinement. The development of rotomoulded (Pop-Iliev et al., 2003) and microcellular foams (Okamato, 2003) are peripheral to the products discussed in this book.

3.3.1 Extrusion of thermoplastic foam sheet The melt, leaving a slot-shaped die, spreads on a moving belt. As the foaming process proceeds, the thermal conductivity of the foam decreases significantly, while the heat conduction distance increases as the foam sheet thickens. The time scale for foam expansion is of the same order as that for heat conduction. Gas diffusion occurs, under the pressure differential from the centre to the surface of the sheet. The lower pressure in the surface cells could allow these to shrink in size and relieve the tensile stresses in the oriented faces. Hence the foam density is expected to be higher near the foam skin. The high tensile viscosity of the molten polymer resists, but does not prevent, the thinning of cell faces and the drawing of melt from cell edges. In the continuous process of PE film blowing, there is a near-equilibrium film thickness in the melt bubble before solidification by crystallisation. The crystallisation rate of LDPE reaches a maximum at around 100°C, and the considerable heat of crystallisation arrests the cooling temporarily. The modelling of foam formation has not yet considered the solidification stage. If neighbouring cells have different diameters, larger bubbles will try to grow at the expense of neighbouring smaller bubbles, lessening the overall surface energy of the system. However, as bubble coarsening is rarely observed, this process must be slow compared with the solidification process. Occasional large cells in foams appear to be the result of face fracture followed by a reshaping of the joined cells. When the melt passes through the short extrusion die, its pressure falls rapidly to atmospheric and a fine cell size is produced. The process of gas diffusion from the melt to the bubbles occurs on a time scale of the order of seconds. Faster pressure reduction allows less time for gas diffusion, so the effective gas diffusion distance is smaller. This allows new bubbles to nucleate closer to growing bubbles, reducing the average cell size in the final foam. Shafi et al. (1996) predicted cell size distributions for freely expanded LDPE/nitrogen system from measured parameters. However they did not predict the mean bubble size in terms of directly measurable process variables.

48 Polymer Foams Handbook 3.3.2 Melt rheology suitable for foaming When low-density foams are produced, the polymer melt undergoes high biaxial extension to form cell faces. Its flow properties (rheology) must suit the process. High molecular weight polymer melts are highly viscous, with part of the deformation being elastic. The melt must sustain high tensile stresses without cell face fracture which would cause neighbouring cells to join. Repeated fracture leads to very large cells, and eventually to foam collapse. PS melts of high molecular weight have a relatively low entanglement density (Mills, 2005), so can undergo high biaxial extensions without fracture. Consequently linear PS can be used for foams. Various techniques have been used to characterise the rheological differences between these polymers. Characterisation of melt elasticity using a small oscillatory shear strain to evaluate the complex shear modulus G* is inappropriate, since it cannot characterise the high strain response. Consequently, tests subject melt extrudates to large tensile deformations. The tensile viscosity is defined by ηT ⬅

σ ε

(3.1)

where σ is the stress and ε. is the tensile strain rate. Ruinaard (2006) was the latest of many researchers who measured the tensile viscosity of polyolefin melts. When a PP melt extrudate was stretched at a constant ε. (Fig. 3.5), the tensile viscosity η increased with time, as the tensile strain increased, due to the entanglement network in the melt

3 4

Viscosity (kPa s)

1000

2

100

1

10 10

1 Time (s)

Figure 3.5

Tensile viscosity as a function of time since stretching started for linear PP1 and branched PP2 to 4 (Ruinaard, 2006 © Soc. Plast. Eng.).


49

becoming significantly extended. The graph shows peak values, after which the melt necks and fails. The graphs for the branched PP2 to 4 extend to a much higher melt viscosities (hence tensile stresses) than that for linear PP1. The breadth of high M side of the MWD was greater for the branched PPs (Fig. 3.6). Similar distributions were found for expanded polypropylene EPP mouldings with MW 艑 260,000 (Mills and Kang, 1994). For linear polymers, the melt elasticity correlates with the ratio (MZ/MW) of molecular weight averages, but branched polymers have increased melt elasticity. The polymer gel content, measured using solvent extraction, is used to characterise the degree of crosslinking of polyolefins. There is a theoretical relationship (Flory, 1953) between these quantities, if the crosslinking occurs at random points. The gel content should not be too high or else polymer flow will be inadequate. As the gel content increases, so does the elastic modulus of the melt, so by the theory in Section 3.2, the density of the foam will also increase. An increased gel content will increase the tensile strength of the molten polymer, so make the foaming process more stable. Crosslinked PE is stable during foaming. Crosslinked metallocene LLDPE (Abe and Yamaguchi, 2001) was characterised in terms of melt moduli C (equation 3.12); the foam relative density increased with the gel fraction. The melt extensional response of the uncrosslinked polymer was unsuitable for foaming (Yamaguchi and Susuki, 2001). However, the addition of 3% of a lightly crosslinked version of the same polymer allowed stable foams to be made. Silane crosslinking of metallocene PE (Pape, 2000) has lower capital cost than conventional peroxide or radiation crosslinking. However the process, it was 1.2 PP1 PP2 PP3 PP4

dW(M)/d(log M)

1.0 0.8 0.6 0.4 0.2 0.0

Figure 3.6

2

3

4

5 log M

6

7

MWD of the PPs shown in Figure 3.5 (Ruinaard, 2006 © Soc. Plast. Eng.).

8

50 Polymer Foams Handbook infeasible until branched metallocene PE was available. The foams can have densities down to 16 kg m⫺3. It is also possible to make open-cell foams from low crystallinity PE copolymers that compete with polyurethane (PU) and PVC open-cell foams. 3.3.2.1

Blowing agents

One challenge has been to use blowing agents, that do not harm the environment, to produce foams with closed, small cells. The chlorofluorocarbons (CFCs) used in the past, such as CFC11 (CCl3F), had several advantages; the heat of fusion of the low boiling point liquids aided foam temperature control, the low diffusivity of the gases made stable cell structures easy to achieve, and the gases were non-flammable. In contrast, pentane and butane are gases at room temperature, flammable, and have a high diffusivity through molten polyolefins. Nevertheless, in the last decade, methods of using pentane and butane have been developed. When a high melt strength PP was foamed using butane (Naguib et al., 2002), the maximum expansion ratio was a function of the extrudate temperature; it increased with temperature in the low temperature range where the expansion was limited by crystallisation, then decreased at higher temperatures due to butane loss from the extrudate. The extrudate swelled from the die, then foaming caused further expansion. Several strategies were used to achieve ultra-low-density PP foams (Naguib and Park, 2000); branched PP prevented cell face fracture, lowering the melt temperature reduced the gas loss during expansion, and optimisation of the die design avoided too-rapid crystallisation. Hydrocerol, a mixture of sodium bicarbonate and citric acid which decomposes to liberate CO2 and a mixture of other products, was used (Behravesh et al., 1996): the CO2 and isobutane acted as blowing agents. The extruder screw speed and hydrocerol concentration controlled the nucleation density, hence the foam density and mean cell size. Branched PP had a slightly larger cell size than linear PP when CO2 was used as the foaming agent, but there were a significant number of open cells in the foamed linear PP (Park and Cheung, 1997). Sims et al. (1997) considered the efficiency of azodicarbonamide and sodium bicarbonate blowing agents for PE foams made by compression moulding. These systems generate CO2 gas. Blends of the blowing agents have a reduced exotherm, so are more suitable for polymer systems with that are temperature sensitive, such as ethylene copolymers. 3.3.2.2

Determining the amount of gas generated

The dimensionless volume Vg0 of gas generated (volume at STP per volume of LDPE) can be calculated from the concentration of the


51

chemical blowing agent. Lee and Flumerfelt (1995) found that the solubility of nitrogen in LDPE melts increases with temperature. The relationship between the mass X of nitrogen dissolved, expressed as g N2/g LDPE, and the pressure p measured in bar, at 135°C is X ⫽ 0.24 ⫻ 10⫺3 p ⫺ 1.7 ⫻ 10⫺6 p2

(for p ⬍ 20)

(3.2)

Since the molar mass of nitrogen is 28 g/mol, and the molar volume of a gas at 135°C is 31,600 ml/mol, 0.22 volumes of nitrogen dissolve in one volume of LDPE at 135°C under an absolute pressure of 1 bar. Since the foam relative densities are typically less than 0.08 in the later stages of expansion, and the gas pressures are less than 0.2 bar, the fraction of the nitrogen gas dissolved in the LDPE is insignificant. It is assumed that no gas loss occurs by diffusion through the cell faces to the outside of the foam. If the foam density is ρ (kg m⫺3) at the process temperature T, the dimensionless gas volume Vg(T,p) under the process conditions is Vg (T ,p ) ⫽

918 ⫺ VP (T ) ρ

(3.3)

The dimensionless LDPE relative volume VP(T) at temperature T °C is given by Hellwege et al. (1962) as VP (T ) ⬅

V (T ) ⫽ 1.057 ⫹ 7.93 ⫻ 10⫺3T V (20)

(3.4)

The absolute gas pressure p is determined from Vg and Vg0 using the ideal gas laws, hence the relative gas pressure pr is obtained. 3.3.2.3

Control of cell size and cell stability

Nucleating agents can be used to reduce cell size. Cheung and Park (1996) described the use of talc in PP foam. Talc is more effective than calcium carbonate (Rodrigue and Gosslin, 2003), probably due to its platelet geometry; the concentration of nuclei appeared to increase almost exponentially with the concentration of talc, with the smallest particle size 0.8 µm talc being most effective. Low-density foams of HDPE can be extruded with fine cells, using CO2 as a blowing agent (Behravesh et al., 1998). The melt temperature was reduced to the lowest possible value of 121°C at the die, to avoid cell coalescence and achieve high expansion ratios. In related research (Lee and Lee, 2000), a blend of LDPE and LLDPE, blown with CO2, was extruded at 220°C. It was necessary to cool the extrudate surface to temperatures as low as 0°C to stabilise the foam.

52 Polymer Foams Handbook 10 Increasing molecular weight Linear polymer (tan δ)

Increased stability

Foam collapse 1

Optimum conditions Gel point Increasing crosslinking

0.1 100

1000

104

105

Complex viscosity (Pa s) Increased resistance to bubble growth

Figure 3.7

Influence of melt viscosity and tan δ, measured at a frequency of 10 Hz, on the stability of LLDPE foam (redrawn from Vachon and Gendron, 2002). For linear polymers of different molecular weights, crosslinking moves the properties in the direction of the large arrow.

Park and Malone (1996) defined a foamability factor F, from the tan δ of the PE melt (at 190°C and 1 Hz), the average cell diameter D, and the foam density ρ as F ⫽ ρ D(tan δ)0.75 ⱕ 1.8

(3.5)

where tan δ is defined as the ratio E⬘/E⬙of the in-phase to the out-ofphase components of the complex Young’s modulus of the melt; a sinusoidally varying shear strain is applied to the melt, and the sinusoidally varying shear stress leads in phase by the angle δ radians. The condition in equation (3.5) is for production of closed-cell foam. The optimum processing window places limits on the melt viscosity and elasticity (Fig. 3.7). If tan δ is too high, so are the tensile stresses when the foam faces extend biaxially. The faces will fracture if the tensile stress exceeds a critical level, causing the formation of abnormally large cells, with more than the usual 14 or 15 faces (Fig. 3.8). Figure 3.7 shows how the crosslinking of a polymer, to just below the gel point, produces the optimum structure.


Figure 3.8

53

Abnormally large cells, with ⬎20 faces, formed by cell face collapse in EVA foam of density 150 kg m⫺3, with a background of normal cells (Verdejo, 2004).

The LDPE blown film process is successful, since the melt bubble cooling occurs in a few seconds, allowing little time for the viscous extensional flow of the thermoplastic melt. However, due to the low thermal diffusivity of foams, it takes the order of 20 min for a PE foam melt to cool to the solid state. During this time, the melt bubbles must remain stable. The typical gel content, from 30% to 70%, causes the low-shear-rate viscosity to be extremely high. However, if gelations were taken further, the tensile stresses in the expanded foam (Section 3.3.3.4) would be too high.

3.3.3 Stages in closed-cell foam development Compared with the five stages of PU foaming, the only two stages in thermoplastics are the growth of isolated spherical bubbles and the formation of polyhedral closed cells. 3.3.3.1

Isolated bubble growth in a melt

The first stage of bubble growth occurs in a polymer melt under pressure, containing dissolved gas. A variety of gases have been used. CFCs have ideal physical properties of low diffusivity, low thermal conductivity, and low boiling point, but have been phased out as they deplete the ozone layer in the stratosphere. Hydrocarbon gases, hydro chlorofluoro carbons (HCFCs), carbon dioxide, and nitrogen are all used.


Polymer Gas S

R or S (µm)

S

1

0.1

R

R

0.01 0.1 (a)

Figure 3.9

(b)

1

10

Foam process time (s)

(a) Modelling the growth of an isolated spherical bubble in a PE melt and (b) the predicted growth in the radii of the bubble and melt with time (redrawn from Koopmans et al., 2000 © Wiley).

A pressure reduction, or the generation of more gas, causes bubbles to nucleate. Figure 3.9a shows an isolated spherical bubble in a polymer melt, the first stage of the foaming process (Fig. 1.3). Shafi et al. (1996) assumed that there are spherically symmetric flows: (a) radial heat flow; (b) radial gas diffusion from the melt to the bubble; (c) extensional polymer melt flow, in directions tangential to the sphere, and compressive melt flow in the radial direction. It is assumed that no further bubble nucleation occurs during bubble growth. The pressure balance of the bubble depends not only on its curvature and surface tension, but also on the biaxial tensile stresses in the melt. In Figure 3.9a, the outer radius of the melt sphere indicates the approximate amount of melt per growing bubble. The viscosity of the polymer melt is a function of its temperature and the amount of dissolved blowing agent. The coupled equations were solved by finite difference methods. Figure 3.9b shows the predicted increase in the cell radius with time in a PS melt. Since the model does not consider the interaction between touching bubbles, it is unlikely to be correct at long times. 3.3.3.2

Multiple bubble growth in a melt

Melt extensional flow in cell faces draws melt from the vertices. Everitt et al. (2006) modelled the interaction of a sheet of large and small bubbles in a 2-dimensional (2D) polymer melt. The bubble


0

0.2

Figure 3.10

0.4

0.5

1

2

55

5

Developing foam in a structure with two large to every small bubble, at the dimensionless times indicated.The grey scale indicates orientation (Everitt et al., 2006).

array had hexagonal symmetry, so it was possible to consider a small representative unit cell (RUC), containing parts of a small and a large bubble plus a mirror symmetry plane. There are separate time scales for polymer viscoelasticity, bubble growth, and for gas diffusion in the melt. Figure 3.10 shows how, with increasing bubble expansion, the melt is elongated between two neighbouring large bubbles. The dimensionless time in the simulation is time divided by the fluid relaxation time. A full 3D consideration of such flow will eventually lead to better models of foam development. 3.3.3.3

The equilibrium density of crosslinked foams

At the end of the face extension process, the fraction φ of the polymer in the cell faces ⬵1. Mahapatro et al. (1998) predicted the density of crosslinked PE foams, expanded by chemically produced nitrogen gas. In the sudden expansion, there is no time for gas loss from the foam surface. The model considers the biaxial extension of the rubbery cell walls, driven by the gas pressure. The Kelvin foam model was used; Figure 3.11 shows the side face of the structural repeat unit (shown in Fig. 7.7). As each square face meets two hexagonal faces at 120° angle, the equal biaxial tensile stresses σf in each face are in equilibrium at the cell edges. It is assumed that the cell face fraction φ is close to 1, so the edge cross-sectional areas can be ignored. The relative pressure pr in the cells is p ⫺ pa, where p is the absolute pressure in the cells, and pa is the atmospheric pressure. A biaxial tensile stress σf acts in a hexagonal cell face of width 冑苳 3L and thickness δ, and in two shared, half square faces, of total width

56 Polymer Foams Handbook L/2 Section through edge

Square face

Gas at pressure pa Hexagonal face of thickness δ

√2L

Gas at pressure pa

Square face 2L

Figure 3.11

Side of Kelvin foam RUC used to calculate the effect of the cell gas pressure on the face tensile stress (redrawn from Mahapatro et al., 1998).

L and thickness δ/2, all of which are perpendicular to the boundary plane shown in Figure 3.11. The total force acting across the boundary must be zero for equilibrium, so pr 2 2L2 ⫺ σf

(

)

3 ⫹ 0.5 Lδ ⫽ 0

(3.6)

This gives the cell face stress in terms of the ratio of the face length L to its thickness δ. Substituting for L/δ, using equation (1.11), gives the stress in terms of the foam relative density R σf ⫽

3pr 2R

(3.7)

The face biaxial extension ratio λ can be estimated by assuming that the cell nuclei form on a body centred cubic (BCC) lattice in the polymer melt, with lattice parameter a. The nuclei separation in the unfoamed melt is δ0 ⫽

a 3 2

(3.8)


57

and the volume of melt per nucleus is a3/2. The melt volume per foamed cell is the product of the cell volume and the relative density, so a3 ⫽ 8 2L3R 2

(3.9)

The foam face expands like a lightly crosslinked rubber, with negligible viscous flow. The thickness extension ratio in the centre of a face, between the unfoamed melt and the foam is λT ≡

δ 2 δ ⫽ δ0 3 a

(3.10)

Substituting for a from equation (3.9) and δ/L from Chapter 1, equation (1.11) gives the result in terms of R. The melt does not change in volume, so the face biaxial tension extension ratio λf is given by λf ⫽

1 λT

⫽

1.703 3

(3.11)

R

The molten crosslinked foam was treated as a rubber, obeying the relationship (Treloar, 1963) between the principal stresses and the extension ratios

(

σ1 ⫺ σ2 ⫽ C ␭12 ⫺ ␭22

)

(3.12)

The constant C, from the kinetic theory of rubber elasticity, is proportional to the density of network chains, and to the absolute temperature. The response of the crosslinked PE was measured in uniaxial tension at some of the lower foaming temperatures. This constant strain-rate data is not ideal, since the lightly crosslinked melt is a viscoelastic material. The data (Fig. 3.12a) fits equation (3.12); the positive offset at zero strain, which may represent the viscous response of the extractable fraction of the polymer, was ignored. 3.3.3.4

Foam density and face tensions

The foam relative density, at the process temperature, is related to its density ρ by Rf ⫽

ρVP (T ) 918

(3.13)


300 10 − 1.2

250

250

150

Stress (kPa)

Stress (kPa)

15 5 − 1.2

200

10 − 0.7

100 5 − 0.7

50 0

0

2

(a)

Figure 3.12

4

6

8

10

Strain (λ2 − λ−1)

12

200 340 Gas 300

150

content (ml g−1)

100

Melt shear modulus (kPa)

50 0

14 (b)

3 0

10

20

30

40

50

60

70

Foam density (kg m−3)

(a) Data for the uniaxial extension of LDPE melt at 140°C, labelled with crosshead speed ⫺ crosslinking agent concentration and (b) graphical solution of the cell wall vs. foam density equations (redrawn from Mahapatro et al., 1998).

The polymer density at 20°C is reduced by the factor VP(T) when heated. Equations (3.2), (3.6), (3.7), and (3.9)–(3.11) can be solved for the gas pressure in the expanded foam, the stress in the cell walls, and the foam density. The solution is at the intersection of a graph representing the gas expansion, and one representing cell face stretching (Fig. 3.12b). The stresses in lightly crosslinked LDPE foam cell faces are of the order of 100 kPa, the biaxial draw ratio is of the order of 3, and the foam density is only slightly reduced from the freeexpansion value (that for zero face stress). Section 3.4 discusses the effects of draw ratio on orientation in LDPE further. The molten cell faces of foamed thermoplastics stress relax before solidification, so the molecular orientation will be less than in crosslinked polymers. Nevertheless the molecular orientation in glassy PS is likely to affect the tensile response and making crazing and fracture less likely. Summary (a) For successful foaming, thermoplastics melts need a high elasticity, so either LDPE, branched PP, or crosslinked polyolefins are used. (b) The polymer in the cell faces is biaxially stretched before it solidifies (crystallises). (c) The cell face thickness profile is determined by the polymer melt flow, not by surface tension.

3.3.4 Post-extrusion shrinkage Once the foam has formed, its geometry must remain stable while the thermoplastic cools and solidifies, and the gases equilibrate. This


(a)

Figure 3.13

59

(b)

LDPE foam of density 14 kg m⫺3 (a) when placed in the SEM and (b) after 64 h under vacuum, when faces have wrinkled (Masso-Moreu and Mills, 2004).

means that the diffusion rate of the gas through the cell faces must be lower than that of air into the foam. As diffusion from foams can be very slow, the product dimensions may change for a long period after manufacture. The Zotefoams process for crosslinked polyolefin foams (Eaves and Witten, 1998) provides the best control of cell diameter, as the nitrogen expansion gas has approximately the same composition as air. The block is cut into sheets when cold, giving full control of the sheet thickness. Nevertheless, the cell faces are likely to be wrinkled in the product (Fig. 3.13). Post-extrusion shrinkage occurs for uncrosslinked PE foam. The diffusion rate of CFCs (used in the past) out of PE foams was lower than the diffusion rate of air into the foam, so the product dimensions were stable. However, pentane or isobutane escapes faster from the foam than air enters, so there is a risk of collapse of the foam dimensions. Yang et al. (2002) modelled the diffusion of gas from LDPE foam of density 22 kg m⫺3, using diffusivities of 1.73 ⫻ 10⫺6 m2 h⫺1 and 0.26 ⫻ 10⫺6 m2 h⫺1 for air and isobutane, respectively. The thickness of LDPE foam shrank by 40% soon after extrusion, followed by a gradual thickness increase that was incomplete after 1 year. However, a glycerol monostearate additive allowed the sheet thickness to become stable after 5 days of storage. When the LDPE was blended with an unspecified ESI interpolymer, this roughly halved the thickness changes, but did not alter their time scale, an effect attributed to the increased polymer Young’s modulus. Yang and Lee (2003) explored the effects of foam density, cell size, and polymer modulus on the rate of

60 Polymer Foams Handbook Table 3.2

Permeabilities and diffusion coefficients for LDPE films

Permeability (barrer) Permeating gas Air n-butane

Diffusion coefficient at 30°C, heat treated (10⫺6 mm2 s⫺1)

LDPE

LDPE ⫹2% SS

LDPE

LDPE ⫹2% SS

1.0 5.3

0.6 0.2

72 4.8

54 1.9

SS: Stearyl stearamide.

diffusion-induced dimensional changes for LDPE foams blown with isobutene. Aging modifiers, such as stearamides and mono-glycerides, are used (Dieckmann and Holtz, 2000) for extruded LDPE foam blown with isobutene. Length shrinkage of the extrudate occurred over a period of about a month. Some of these modifiers have anti-static properties, important when a flammable gas is used as the blowing agent. A distilled mono-glyceride was the most effective at stabilising the dimensions. Bouma et al. (1997) considered a range of alkane blowing agents, and Nauta (2000) provides details of the science. Table 3.2 gives the permeabilities of LDPE films measured in barrer ⫽ 10⫺10 cm3(STP) cm/ (cm2 s cmHg). Films were used rather than foam, since the polymer geometry is not accurately known in the latter. However the orientation of the polymer crystals in the film may differ from those in foam faces. The permeability of n-butane through LDPE film is 5 times that of air; however it is 1/3 of the air value when stearyl stearamide is added, forming a film on the surface of the LDPE. In LDPE foams, stearyl stearamide migrates to the cell face surfaces. Heat treatment of the LDPE film (1 h at 83°C) reduces its diffusivity to butene, but not to air. It hardly changes the solubility of butene in the PE. Nauta (2000) measured volume changes of a 5 mm thick foam sheet of density 28 kg m⫺3, with cell diameter 0.45 mm and face thickness 4 µm (Fig. 3.14). Modelling could successfully reproduce the experimental data. The volume contraction due to butene loss occurs in less than 1 h, while the expansion due to air ingress is incomplete after 3 days. When PE is extruded as 75 mm or thicker planks, the processes are slower. Assuming that Fick’s law applies, the time, for a certain % gas loss, increases with the square of the foam sheet thickness. The gas permeability of PEs decreases, as the crystallinity increases, due to increases in the tortuosity of the gas path through, and restriction of the mobility of, the amorphous phase.


61

1 No SS

Relative volume

0.9 0.8

1.5% SS

0.7 0.6 0.5 0.4

Figure 3.14

0

10

20

30 Time (h)

40

50

Volume changes due to diffusion of butene from, and air into, a 5 mm thick foam sheet of density 28 kg m⫺3, with and without stearyl stearamide additive (redrawn from Nauta, 2000). SS: Stearyl stearamide.

3.3.5 Oriented PP foams – Strandfoam Dow Strandfoam is a PP foam with oriented elongated cells as in wood. A high melt strength PP is extruded from a die having 722 holes of diameter 1 mm (Park and Garcia, 2000). The blowing agent must be highly soluble in the melt, yet have a low permeability at room temperature. A 60/40 mixture of an HCFC (having a much lower permeability in PP than air) and ethylene chloride (much higher permeability) achieved optimal dimensional stability of extruded plank. Low foam densities (typically 20 kg m⫺3) and small cell diameters (typically 0.9 mm) were required to produce foams with ⬎80% closed cells. The bundle of extrudates are fused together, by low-friction plates which press on their sides and consolidate them into a hexagonal array. Slices, cut across the extrudate, have the strong direction normal to the slice surface, which is optimal for the cores of sandwich structures; their role is to keep the skins at constant separation. They also have a high shear modulus, preventing easy shear of the sandwich structure (Chapter 18). Strandfoam competes with paper or metal honeycombs in this application. The cell axis orientation in Strandfoam is constant, so it is unsuitable for curved products such as helmet liners, which require the high yield stress direction to vary. In automobiles it is used for occupant protection; complex shapes can be sawn from block, cut by abrasive wires, or thermoformed.


3.4

Foam crystallinity and crystal orientation The concept of percentage crystallinity implies a two-phase structure, of perfect crystals in a matrix of amorphous material. As defects exist in the crystals, and inter-crystalline links cross the amorphous regions, the two-phase structure is an approximation. The various techniques of measuring the average crystallinity of polymers depend on different physical properties, so give slightly different results. Measurements of density or latent heat of fusion are the easiest to perform. Mills (1997) used a differential scanning calorimeter (DSC) to determine the heat of fusion of a PP, used in a foam, as 67 J g⫺1. Since isotactic PP has a heat of fusion of 209 J g⫺1, the volume fraction crystallinity is 0.30. If the foam can be remoulded into a solid, assuming that this process does not change the crystallinity, polymer density measurement can be used to determine crystallinity. The PP crystal and amorphous densities are 936 and 853 kg m⫺3, respectively, at 20°C. The density of a solid disc, moulded from the PP foam mentioned above, was 889 kg m⫺3 at 20°C. This implies a volume fraction crystallinity of 0.43, somewhat higher than the DSC value. Both the low crystallinity and the low melting point show that the PP is a copolymer. Rodriguez-Perez et al. (2005) used Raman spectroscopy to determine the local values of crystallinity in the edges, vertices, and faces of PE foams from the Zotefoams process (Fig. 3.15). Scanning electron microscope (SEM) of etched cell faces (Fig. 3.16) shows disclike structures, while there are spherulites in the cell edges. X-ray diffraction can in principal reveal the crystal orientation in the foam structural elements, but, to date, no such data has been published. In 70.0 60.0

Ic (%)

50.0 HD30

40.0

HL34 LD29

30.0

VA25 20.0 10.0 0.0 Edge Vertex Wall perpendicular perpendicular perpendicular

Figure 3.15

Edge in plane

Wall in plane

Solid sheet

Crystallinity of cell features from Raman (Rodriguez-Perez et al., 2005 © Elsevier). Key: HD, high density; HL, high/low density; LD, low density; VA, vinyl acetate copolymer; 30 etc: foam densities (kg m⫺3).


63

blown PE film, X-ray pole figures have been used to characterise the crystal orientation, but it is not possible to collect a large number of foam faces, and orient them identically, prior to X-ray examination. The melting range of the foam, measured by DSC, can be interpreted as a distribution of lamellar thicknesses. Figure 3.17 shows such data for Zotefoam LDPE foams. Biaxial molecular orientation increases the in-plane strength of cell faces, compared with the polymer in the bulk state. For instance, PS crazes when a high tensile stress is applied to an injection moulded product that is millimetres thick, but biaxially oriented PS films yield in tension (Fig. 3.3). ∼20µm

∼20µm

Figure 3.16

SEM of etched cell faces of LDPE foam (Almanza et al., 2005 © Elsevier).

0.30

×103 (1/M)dM/dL

0.25 LD24 LD33 LD29 LD60

0.20 0.15 0.10 0.05 0.00 0

25

50

75

100

L (Å)

Figure 3.17

Lamella thickness distribution in LDPE foams of density (kg m⫺3) indicated (Almanza et al. 2005 © Elsevier).


Summary The geometry of foam cells needs to be quantified to assist the modelling of mechanical properties (Chapter 11). Statistical descriptions are necessary because of the variability in the structure. In closed-cell thermoplastic foams, the faces, which contain the great majority of the polymer, dominate the mechanical response. The faces may be wrinkled as a result of post-foaming shrinkage.

References Abe S. & Yamaguchi M. (2001) Study on the foaming of crosslinked polyethylene, J. Appl. Polym. Sci. 79, 2146–2155. Almanza O., Rodriguez-Perez M.A. et al. (2005) Comparative study on the lamellar structure of polyethylene foams, Euro. Polym. J. 41, 599–609. Ankrah S., Verdejo R. & Mills N.J. (2002) The mechanical properties of ESI/LDPE foam blends and sport applications, Cell. Polym. 21, 237–264. Behravesh A.H. et al., (1998) Challenge to the production of lowdensity fine-cell HDPE foams using carbon dioxide. Cell. Poly. 17, 309–326. Behravesh A.H., Park C.B. et al. (1996) Extrusion of polypropylene foams with hydrocerol and isopentane, J. Vinyl Addit. Tech. 2, 349–357. Benning C.J. (1967) Part I: Modified PE foam systems. J. Cell. Plast. 3, 62–72; Part II: Mechanical props of polyethylene foams prepared at high and low expansion rate, 125–137; Part III: Orientation in thermoplastic foams, 174–184. Bistac S., Vallat M.F. & Schultz J. (1999) Study of ethylene copolymer films by dielectric spectroscopy, Prog. Org. Coat. 37, 49–56. Bouma R.H.B., Nauta W.J. et al. (1997) Foam stability related to polymer permeability: 1 low MW additives, J. Appl. Polym. Sci. 65, 2679–2689. Chaudhary B.I., Barry R.P. & Tusim M.H. (2000) Foams made from blends of ESI with PE, PP and PS, J. Cell. Plast. 36, 397–421. Cheung L.K. & Park C.B. (1996) Effect of talc on the cell population density of extruded polypropylene foams, in Cellular and Microcellular Materials, MD- Vol. 76, ASTM. Dieckmann D. & Holtz B. (2000) Ageing modifiers for extruded LDPE foam, J. Vinyl Addit. Tech. 6, 34–38. Eaves D.E. (2001) Polymer foams: trends in use and technology, RAPRA Industry Analysis Report. Eaves D.E. & Witten N. (1998) Product and process development in Zotefoams polyolefin foam manufacture, ANTEC Conference.


65

Everitt S.L., Harlen O.G. & Wilson H.J. (2006) Competition and interaction of polydisperse bubbles in polymer foams, J. Non-Newt. Fluid Mech. 137, 60–71. Flory P.J. (1953) Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY. Folland R., Reichelt N. et al. (2002) Low density extruded polypropylene foams, Foams 2002 Conference, Houston, Session I, pp. 1–8. Gendron R., Ed. (2005) Thermoplastic Foam Processing: Principles and Development, CRC Press, Boca Raton, FL. Gibson L.J. & Ashby M.F. (1988) Cellular Solids, 2nd edn., Cambridge University Press, Pergamon, Oxford. Hellwege K.H., Knappe W. & Lehmann P. (1962) Kolloid-Zu. Z. Polymere, 183, 110–120. Kontopoulou M., Huang L.C. & Lee J.A. (2003) Binary blends of EVA and metallocene α olefin copolymers and their film props, Adv. Polym. Tech. 22, 209–217. Koopmans R.J., den Doelder J.C. & Paquet A.N. (2000) Modelling foam growth in thermoplastics, Adv. Mater. 12, 1873–1880. Kristen M.O. (1999) Metallocenes – from a laboratory curiosity to industrial scale applications, on academic.sun.ac.za/unesco/ polymerED2000/conf1999/Lectures1999/KRISTEN.pdf Krusche A. & Haberstroh E. (1999) Optimization of the production of EPDM sponge rubber seals for the automotive industry, 155th ACS Rubber Division Meeting, Chicago, IL, Paper 69, p. 36. Lee J.G. & Flumerfelt R.W. (1995) Nitrogen solubilities in low-density polyethylene at high temperatures and high pressures, J. Appl. Polym. Sci. 58, 2213–2219. Lee S.-T. & Lee K. (2000) Surrounding temperature effects on extruded polyethylene foam structure, Adv. Polym. Tech. 19, 87–96. Liu I.-C. & Tsiang R.C.-C. (2003) Tailoring viscoelastic and mechanical properties of the foamed blends of EVA and various ethylene–styrene interpolymers, Polym. Compos. 24, 304–313. Mahapatro A., Mills N.J. & Sims G.L.A. (1998) Experiments and modelling of the expansion of crosslinked polyethylene foams, Cell. Polym. 17, 252–270. Masso-Moreu Y. & Mills N.J. (2004) Rapid hydrostatic compression of low density polymeric foams, Polym. Test. 23, 313–322. Mills N.J. (1997) Time dependence of the compressive response of polypropylene bead foam, Cell. Polym. 16, 194–215. Mills N.J. (2004) Polyolefin Foams, Review Report 167, Vol. 14, no. 11, RAPRA Ltd, Shawbury, Shropshire. Mills N.J. (2005) Plastics–Microstructure and Applications, 3rd edn., Butterworth, London. Mills N.J. & Kang P. (1994) The effect of water immersion on the fracture toughness of polystyrene foam used in soft shell cycle helmets, J. Cell. Plast. 30, 196–222.

66 Polymer Foams Handbook Mills N.J. & Zhu H. (1999) The high strain compression of closed-cell polymer foams, J. Mech. Phys. Solid. 47, 669–695. Naguib H.E. & Park C.B. (2000) Challenge to the production of low density PP foams by extrusion, Foams 2000, 149, SPE. Naguib H.E., Park C.B. et al. (2002) Strategies for achieving ultralow density PP foams, Polym. Eng. Sci. 42, 1481–1492. Nauta A. (2000) Stabilisation of low density, closed cell polyethylene foam, Ph.D. thesis, at www.ub.twente.nl/webdocs/ct/1/t00001c.pdf Okamato K.T. (2003) Microcellular Processing, Carl Hanser, Munich. Pape P.G. (2000) Moisture crosslinking process for foamed polymers, J. Vinyl Addit. Tech. 6, 49–52. Park C.P. (2004) Polyolefin foams, Chapter 9 in Polymeric Foams and Foam Technology, 2nd edn., Eds. Klempner D. & Sendijarevic V., Hanser, Munich. Park C.P. & Cheung L.K. (1997) A study of cell nucleation in the extrusion of PP foams, Poly. Eng. Sci. 37, 1–10. Park C.P. & Garcia G.A. (2000) Development of polypropylene plank foam products, ANTEC 2000 Conference, Paper 404. Park C.P. & Malone B.A. (1996) Extruded, closed-cell polypropylene foam, US Patent 5527573. Phillips E.M., McHugh K.E. & Bradley M.B. (1992) PP with high melt stability, Kunststoffe 82, 671–676. Pop-Iliev R., Liu F. et al. (2003) Rotational foam molding of polypropylene with control of melt strength, Adv. Polym. Tech. 22, 280–296. Ramesh N.S. (2000) New products made from ethylene–styrene interpolymer/IDPE blend foams for various applications, Foams 2000 Conference, pp. 102–107. Rodrigue D. & Gosselin G. (2003) The effect of nucleating agents on PP foam morphology, Blowing Agents and Foaming Processes conference, Munich, RAPRA Technology, Shawbury. Rodriguez-Perez M.A., Duijsens A. & de Saja J.A. (1998) Effect of addition of EVA on properties of extruded foam profiles of LDPE EVA blends, J. Appl. Polym. Sci. 68, 1237–1244. Rodriguez-Perez M.A., Campo-Arnaiz R.A. et al. (2005) Characterisation of the matrix polymer morphology of polyolefin foams by Raman spectroscopy, Polymer 46, 12093–12102. Ruinaard H. (2006) Elongational viscosity as a tool to predict the foamability of polyolefins, J. Cell. Plast. 42, 207–220. Shafi M.A., Lee J.G. & Flumerfelt R.W. (1996) Prediction of cellular structure in free expansion polymer foam processing, Poly. Eng. Sci. 36, 1950–1959. Sims G.L.A. & Sirithongtaworn W. (1997) Azocarbonamide and sodium bicarbonate blends as blowing agents for crosslinked PE foams, Cell. Polym. 16, 271–283. Thoen J., Zhao J. et al. (2002) New development in INSPIRE performance polymers for foams, Foams 2002 Conference, 5.


67

Treloar L.R.G. (1963) The Physics of Rubberlike Elasticity, Oxford University Press. Vachon C. & Gendron R. (2002) Effect of viscosity on low density foaming of poly (ethylene-co-octene) resins. ANTEC 2002 Conference. Verdejo R. (2004) Gas loss and durability of EVA foams used in running shoe soles, Ph.D. thesis, University of Birmingham. Yamaguchi M. & Susuki K.-I. (2001) Rheological properties and foam processability for blends of linear and crosslinked polyethylenes, J. Polym. Sci. Polym. Phys. 39, 2159–2167. Yang C.T. & Lee S.T. (2003) Dimensional stability of foams based on LDPE ESI blends, J. Cell. Plast. 39, 59–69. Yang C.T., Lee K.L. & Lee S.T. (2002) Dimensional stability of LDPE foams: model and experiments, J. Cell. Plast. 38, 113–128.

Chapter 4

Bead foam microstructure and processing

Chapter contents 4.1 Introduction 4.2 Processing 4.3 Microstructure 4.4 Specific bead foams References

70 70 74 80 82


4.1

Introduction Bead foams, based on commodity thermoplastics (Table 4.1), are widely used for packaging (Chapter 12), thermal insulation, and in some civil engineering applications. Bead moulding allows the economic massproduction of complex shaped products, with high dimensional accuracy. The process and the foam microstructure were reviewed by Svec et al. (1990). In the lost-foam or expendable pattern casting process (Plwonka, 1990), molten aluminium replaces the expanded polystyrene (EPS) pattern as it enters a sand mould.

4.2

Processing 4.2.1 Bead preparation Pentane is introduced to suspension-polymerised polystyrene (PS) beads. Molecular weights in the range of 150,000–250,000 are used (see Section 4.4.2). The spherical beads are sieved to a narrow range of diameters, so are not monodisperse. PS beads are allowed to absorb up to 8% by weight of pentane, which is a plasticiser, reducing the glass transition temperature from 100°C to about 60°C (Fig. 4.1a). Cigna et al. (1986) used an ultramicrotome to section the pentanecontaining beads, and then transmission electron microscopy. Microvoids within them (Fig. 4.1b) are the sites of the later cell expansion. The surfaces of the beads are coated, typically with less than 0.5% of calcium stearate, to prevent agglomeration. Expanded polypropylene (EPP) goes through a process of extrusion foaming as small diameter rod, then pelletised into beads, much as most thermoplastics are pelletised prior to injection moulding. Hence the initial production is as described in the last chapter, and high melt strength polypropylene (PP) grades must be used. The blowing agent is probably isopentane or CO2. Some of the beads have a solid PP outer layer, and their average density is lower than the final EPP mouldings (Beverte, 2004), a disadvantage compared with EPS in that bulky material must be transported to the site of moulding.

Table 4.1

Polymers used for bead foams Polymer name

Abbreviation

Type

Polyethylene Polypropylene Polystyrene PS/PE interpolymer

EPE EPP EPS Arcel

Semi-cryst Semi-cryst Glass

Chapter 4 Bead foam microstructure and processing

71

Glass transition temperature (°C)

100

90

80

70

60 (a)

Figure 4.1

0

2

4 6 Pentane weight fraction (%)

8 (b)

(a) Effect of pentane content on the glass transition temperature of PS (redrawn from Fen-Chong et al., 1999b © Sage) and (b) microvoids seen in an ultramicrotomed section of a bead (Cigna et al., 1986 © Wiley).

4.2.2 Steam moulding When PS beads are heated in the first stage of the process, the pentane begins to evaporate at about 50°C. If the beads are heated to 110°C, their density decreases for the first 5 min. Figure 4.2 shows the diameter expansion of individual beads at different temperatures. At longer times the pentane escapes faster than air diffuses into the cells, and the cells begin to collapse. Figure 4.3 shows the microstructure change during the expansion; in Figure 4.3a, the beads have become white (suggesting many microvoids) while in Figure 4.3b, the expansion is greater and near spherical voids have formed. There are skins on the surfaces of the pre-expanded beads. The beads are then cooled and allowed to mature for about a day. During this time air diffuses into the cells and pentane is lost. Proprietary surfactants are used on the bead surfaces, to control the bead fusion process. An aggressive surfactant can attack bead surfaces, making them rougher, and can even cause surface cell faces to fracture. In the second stage, sufficient beads are blown into a mould (Fig. 4.4) to almost fill it. Wall et al. (2003) discussed control of the mass of injected beads, which is important to produce mouldings of consistent size. The bead size is chosen so the product has at least three beads through its thickness – this is particularly important for EPS coffee cups with wall thicknesses of about 3 mm; if large beads are used, the cups may leak. Steam is then passed through ports in the mould wall at a pressure in the range 1.29–1.98 bar. This corresponds to temperatures between

72 Polymer Foams Handbook 2 Relative diameter expansion

100 1.5 86 1 80

0.5 76 0

Figure 4.2

(a)

Figure 4.3

72 0

5

10 Time (min)

15

20

Relative diameter expansion of beads containing 8% pentane at temperatures (°C) shown (redrawn from Fen-Chong et al., 1999b © Wiley).

(b)

Bead microstructure during expansion: (a) cloudy phase at 95°C and (b) bubbles at 115°C (Tuladhar and Mackley, 2004 © Elsevier).

107°C and 120°C, above the Tg of the pentane-containing PS. When EPP is moulded higher steam pressures (3.5 bar) are used (Nakai et al., 2006). The flow direction of the steam through the mould may be periodically reversed, to prevent the build-up of condensed water. The partial steam pressure inside the beads is zero before moulding. As the rate of steam diffusion into the beads is higher than that of air or pentane out of the beads, the internal pressure increases, causing a slight expansion. Consequently flat regions develop between the beads,

Chapter 4 Bead foam microstructure and processing Steam

73

Steam

Air ⫹ water Moulding

Steam vent Air ⫹ beads Movable mould half

Air ⫹ water Air ⫹ water or vacuum

Figure 4.4

Cross-section of a bead foam mould showing the entry points for beads, steam, and cooling water.

and a welding process starts. However, if beads are steamed for too long, they start to collapse (Stupak et al., 1991). When the steam is cut off and cooling water applied to the mould, the mould must remain closed until the internal pressure falls to zero. The cooling rate of a 25 cm side EPS cube (Skinner and Eagleton, 1964) was either normal or anomalous rapid. In the latter case there was a negligible temperature gradient from the centre to the surface of the block, but there were more holes and voids at the bead boundaries. The mould pressure fell to zero when the temperature, in a foam of density of 32 kg m⫺3, fell to 90°C. Higher-than-expected foam cooling rates occur if there is easy steam transfer out of the moulding via open channels at the bead boundaries. EPS moulders use conditions that allow rapid cooling, unless there is a stringent mechanical property requirement on the foam. Additives, such as 0.1% hexachlorobromododecane, produce a moulding microstructure that allows rapid cooling (Svec et al., 1990). Scanning electron microscope (SEM) of fracture surfaces reveals the small channels where three beads meet (Fig. 4.5). These channels aid heat transfer during moulding, hence reduce the cycle time. The time available for the fusion of the bead surfaces is between 10 and 100 s, depending on position in the moulding – the outer layers solidify before the interior.


0.5 mm

Figure 4.5

Channels (arrowed) at the bead boundaries of moulded EPS foam (Mills and Kang, 1994).

4.2.3 Dimensional stability post-moulding Dimensional stability is essential for the lost-foam metal casting process, but less so for packaging. EPS typically shrinks 0.6% in length, over a period of 24 h at 20°C (Jarvela et al., 1986b), who also studied the larger shrinkage at 60°C. As this temperature is rather close to the polymer Tg, orientation relaxation in the cell faces probably contributed to the shrinkage. Fen-Chong et al. (1999a) modelled the after-shrinkage of EPS in terms of the viscoelastic response of the PS. The diffusion of gases from the foam cells must also play a part, as it does for polyethylene (PE) foams (Chapter 3). This diffusion is only to the inter-bead channels, which are connected to the external air.

4.3

Microstructure 4.3.1 Bead shape and fusion When the spherical beads in the mould are steamed, they expand and become approximately polyhedral. Frequently the fracture surface of the foam runs along bead boundaries (Fig. 4.6). Thus the bead shapes are similar to the cell shapes in ‘wet’ foams, but on a larger scale. There can be open air-filled channels between three beads. The shape of these


Figure 4.6

75

Flat bead faces and polyhedral bead shapes on the fracture surface of EPS foam with a low degree of fusion (Mills, unpublished).

is identical to that of polymer edges in open-cell foams, but again on a larger scale. The channels can connect into a continuous network, which then assists the long-distance flow of steam and other gases during foam moulding. The degree of fusion is assessed by visual inspection of SEM micrographs of fracture surfaces; Rossacci and Shivkumar (2003) show typical appearances over the range (Fig. 4.7). Chapter 15 describes how the crack path in EPS mouldings may follow the bead boundaries.

4.3.2 Density variations in large mouldings Skinner et al. (1964) found a twofold variation in density, in foams of nominal density 16 kg m⫺3, but less variations in foams of nominal density ⬎32 kg m⫺3. Such density variation affects the analysis of flexural strength tests (see Chapter 5). Figure 4.8 shows the density variation across a 50 mm thick moulding of PS bead foam (Moosa and Mills, 1998).

4.3.3 The effects of processing on properties Jarvela et al. (1986a) studied the development of tensile strength at the boundary between two beads as a function of the steam pressure


Figure 4.7

(a)

(b)

(c)

(d)

(e)

(f)

Range of fracture surface appearance from bead boundary to trans-bead. The markers are 400 µm (Rossacci and Shivkumar, 2003 © Kluwer).

and contact time. The strength increase was non-linear with time and slightly faster at higher pressures (Fig. 4.9). The tensile failure stress of foam mouldings of density 22 kg m⫺3 increases almost linearly with the degree of boundary fusion (Fig. 4.10). Stupak et al. (1991) measured the toughness J of EPS mouldings, using 6 moulding times and 6 steam pressures. In their Latin Square experimental design, for a 23 kg m⫺3 foam density, the shortest moulding time went with the highest pressure, and so on. Although they drew


77

110

Density (kg m−3)

90

A-80

B-50

B-80

A-50

70

50

30

Figure 4.8

0

10

20 30 Distance from top of tile (mm)

40

50

Density variation through the thickness of EPS foam tiles of nominal densities 50 and 80 kg m⫺3 (Moosa and Mills, 1998).

1.5

Tensile strength (MPa)

1.5 0.9 1.1 0.3 1.25

1

0.75

Figure 4.9

0

20

40 Time (s)

60

80

Tensile strength vs. steaming time at the steam pressures (bar) marked (redrawn from Jarvela et al., 1986a © Kluwer).

trend lines of J increasing nearly linearly with moulding time, there was no data to support these; the more likely variation would be that in Figure 4.9. In Figure 4.11 the apparent decrease in toughness with moulding time, results from the decrease in the steam temperature.


Tensile strength (MPa)

0.5 0.4 0.3 0.2 0.1 0

Figure 4.10

0

20

40 60 Degree of fusion (%)

80

100

EPS of density 22 kg m⫺3: variation of tensile strength with degree of fusion (Rossacci and Shivkumar, 2003 © Kluwer).

Toughness J (J m⫺2)

120 118

100

115

80 120

113

60

110

40 20

0

50

100

150

107

200

250

Moulding time (s)

Figure 4.11

Toughness of EPS mouldings of density 23 kg m⫺3 vs. moulding time, at the steam temperatures (°C) indicated (data from Stupak et al., 1991).

Figure 15.15 shows that the toughness J increases with the proportion of trans-bead fracture. Development of tensile strength between the beads involves the diffusion of molecules across the interface and their subsequent entanglement. Kline and Wool (1988) found the shear strength of a welded joint between molten PS surfaces increased with the 0.25th power of time. The process is slower for higher molecular weight polymers, but faster at higher temperatures. The EPS molecular weight is high compared with the entanglement molecular weight of 35,000, so it may


(a)

Figure 4.12

79

(b)

(a) Section through EPS moulding, with moulded surface at left and (b) surface appearance.The dark dots are from the steam entry holes (Mills, unpublished).

be able to form strong welds during the time that the bead surfaces remain above Tg. Surface beads in a moulding have a lower welding time, since they cool faster.

4.3.4 Bead shape variation The bead size, shape, and density can vary with position in the moulded product. Figure 4.12a shows a section through an EPS moulding. The beads near the surface have not deformed sufficiently to eliminate the inter-bead channels. Most beads are distorted spheres, with flat patches in contact with neighbouring beads. On the surface of such mouldings, the bead boundaries can be slightly curved (Fig. 4.12b). If a PS bead is at a higher pressure than its neighbour, the pressure differential causes the inter-bead boundary to be curved. Hence curved boundaries indicate either that the beads contain differing amounts of blowing agent or that they have expanded by differing amounts. There is likely to be a density variation from moulded bead to moulded bead, since the space for expansion is variable, and beads cannot move relative to their neighbours once they begin to fuse at the boundaries. The cell shapes inside a moulded bead can vary. The bead has a solid skin, which takes a near spherical shape. Cells close to this tend to have brick-like shapes, with two of their faces parallel to the bead boundary. Cells in the interior of the bead have equiaxed polygonal shapes. A skin-core morphology variation can influence the mechanical properties of the moulded foam, as the denser skins are of higher modulus.

80 Polymer Foams Handbook 4.3.5 Microstructural models The microstructural rules, that apply to cells in wet polyurethane (PU) foams (Section 1.7), also apply to the beads in bead foams. The shape of the edge plus vertex structure in such PU foams (Fig. 1.2(d)) is similar to the shape of the channel network in bead foams. Chapter 11 examines how this affects the mechanical properties. In bead mouldings, there are likely to be: (a) channels for the passage of gases or liquids; (b) greater density near the surface than the core; (c) variations in bead shape, with flattened beads near the surface.

4.4

Specific bead foams 4.4.1 PP bead foam: EPP BASF ‘Neopolen P’ foam blocks are made with a range of densities (Table 4.2). Mills (1997) used image analysis to detect the largest diameter of each bead on the exterior surface of moulded blocks. The average surface bead diameter was only slightly larger in the 20 kg m⫺3 than in the 60 kg m⫺3 density foam, suggesting that the un-expanded beads differed in size. Internal bead diameters, measured on a section perpendicular to the surface, were similar to the external diameters, except within 5 mm of the moulded surface, where the beads were flattened. The volume fraction of PP in the cell faces was calculated assuming a Kelvin foam structure with plateau border edges and ignoring the vertex volumes. The radial distance r was measured for a number of – sectioned edges and the mean – value r computed. In the Kelvin foam, the edge length L ⫽ D/2冪2, where D is the cell diameter. The same relationship was assumed to apply between the average edge length – – L and the mean cell diameter D in the real foam. Assuming that equations (1.3) and (1.7) apply to the mean edge radius –r , the edge relative

Table 4.2

PP foam densities, bead, and cell sizes Nominal density (kg m⫺3) 20 30 43 60

Measured density (kg m⫺3) 29.7 60.1

Mean bead diameter (mm)

Mean cell diameter – D (mm)

4.7 ⫾ 1.2 3.5 ⫾ 1.2 4.3 ⫾ 0.9 3.8 ⫾ 0.8

0.26 ⫾ 0.12 0.23 ⫾ 0.10 0.23 ⫾ 0.10 0.17 ⫾ 0.09

Mean edge radius –r (µm) 2.8 3.1 3.5 ⬍6.9

Face volume fraction φ 0.72 0.71 0.74 艑0.5


81

– density is Re ⫽ 53.9(r–/D)2 Knowing the foam relative density, the volume fraction of polymer in the faces, φ can be computed. Table 4.2 shows that it is high for all but the highest density PP foam, but not as high as for PS bead foams.

4.4.2 PS bead foam: EPS Mills and Kang (1994) sampled commercial mouldings (Table 4.3). The degree of fusion between bead surfaces, revealed by inspection of fracture surface appearance, was high for the cycle helmet mouldings but lower for the other mouldings. The bead boundary contrast on the moulding surface was enhanced by immersion in black ink for several hours (Fig. 4.13). This showed that a crack followed bead boundaries in this case.

Table 4.3

Figure 4.13

PS bead foams

Source

Form

Cycle helmet Cycle helmet Box lid Box lid Packaging

Curved Curved Flat Flat Box

Thickness (mm)

Mean density (kg m⫺3)

Mean bead diameter (mm)

⬍30 ⬍30 20 20 30

50.4 70.4 56.5 21.7 20.0

4.15 ⫾ 1.26 2.56 ⫾ 0.78 2.35 ⫾ 0.63 3.04 ⫾ 0.78 2.78 ⫾ 0.85

Crack path, from a cut crack, along the bead boundaries on an EPS foam moulded surface (Mills and Kang, 1994).


MWD of EPS foams Country of manufacture UK USA USA

MW (1000)

MW/MN

220 283 236

1.70 2.06 2.01

If the beads were uniform sized spheres, the average diameter on a cut section would be 86.6% of the real diameter. The average diameter on an internal section, for the first foam in Table 4.3, was 75% of the surface measurement. This suggests that there is some sideways expansion of the surface beads. The beads are flattened into the moulded surface plane, but the bead shapes are equiaxed in the core of the moulding. Away from the moulded surface layer, there were no significant variations in the mean bead diameter. Since EPS expands at a low temperature compared with that used for PS foam extrusion, the melt viscosity will be very high, and the level of melt elasticity will not be critical for cell stability. However, the molecular weight affects the ease of diffusive bonding of bead boundaries. Table 4.4 shows the molecular weight distribution (MWD) of typical foams, measured by gel permeation chromatography (GPC) by Mills and Kang (1994). The MW of EPS is higher than that of injection moulding grades, but the MWD width, defined as the ratio of MW to the number average molecular weight MN, is approximately 2, as expected for random termination of an addition polymer. Cigna et al. (1986) also determined MW/MN ⫽ 2 for PS foams. This value compares with 5.6 for an Edistar injection moulding grade showing that PS for bulk mouldings is made by a different process.

References Beverte I. (2004) Deformation of polypropylene foam Neopolen P in compression, J. Cell. Plast. 40, 191–204. Cigna G., Merlotti M. & Castellani L. (1986) Morphological and kinetic study of EPS pre-expansion and effects on foam properties, Cell. Polym. 5, 241–268. Fen-Chong T., Herve E. & Zaoui A. (1999a) Micromechanical modelling of intracellular pressure induced viscoelastic shrinkage of EPS foam, Eur. J. Mech. A Solid. 18, 201–218. Fen-Chong T., Herve E. et al. (1999b) Viscoelastic characteristics of pentane-swollen polystyrene beads, J. Appl. Polym. Sci. 73, 2463–2472.


83

Jarvela P., Sarlin P. et al. (1986a) A method to measure the fusion strength of EPS beads, J. Mater. Sci. 21, 3139–3142. Jarvela P., Pohjonen T. et al. (1986b) The after-shrinkage of EPS and a method to eliminate it at the working temperature range, Cell. Polym. 5, 289–301. Kline D.B. & Wool P. (1988) Polymer welding relations investigated by a lap shear joint method, Poly. Eng. Sci. 28, 52–57. Mills N.J. (1997) Time dependence of the compressive response of polypropylene bead foam, Cell. Polym. 16, 194–215. Mills N.J. & Kang P. (1994) The effect of water immersion on the fracture toughness of polystyrene foam used in soft shell cycle helmets, J. Cell. Plast. 30, 196–222. Moosa A. & Mills N.J. (1998) Analysis of bend tests on polystyrene bead foams, Polym. Test. 17, 357–378. Nakai S., Taki K. et al. (2006) Numerical simulation of stress on the mold in bead expansion process, ANTEC 2006, 2726–2730. Plwonka T.S. (1990) A comparison of lost pattern casting processes, Mater. Des. 11, 283–290. Rossacci J. & Shivkumar S. (2003) Bead fusion in polystyrene foams, J. Mater. Sci. 38, 201–206. Skinner S.J. & Eagleton S.D. (1964) Some aspects of the steam moulding process for expandable polystyrene, Plast. Inst. Trans. 32, 180–187, 212–216, 231–239. Stupak P.R., Frye W.O. & Donovan J.A. (1991) The effect of bead fusion on the energy absorption of polystyrene foam, Part I. Fracture toughness, J. Cell. Plast. 27, 484–505. Svec P., Rosik L. et al. (1990) Styrene Based Plastics and Their Modification, Ellis Horwood, London. Tuladhar T.R. & Mackley M.R. (2004) Experimental observations and modelling related to foaming and bubble growth from pentane loaded polystyrene melts, Chem. Eng. Sci. 59, 5997–6014. Wall K.F., Bhavnani S.H. et al. (2003) Investigation of the performance of an expandable polystyrene injector for use in the lost foam process, Met. Mat. Trans. 34B, 843–851.

Chapter 5

Simple mechanical tests

Chapter contents 5.1 Introduction 5.2 Stiffness and strength of structures 5.3 Stress–strain responses and material parameters 5.4 Test types 5.5 Testing products with a density gradient 5.6 Test equipment References

86 87 88 93 105 107 113


5.1

Introduction This chapter considers compressive, tensile, and shear tests on foams and their interpretation. The time scale varies from days for creep tests, to a few milliseconds for impact tests. Specialised tests, such as indentation force deflection tests for soft foams and fracture toughness tests for rigid foams, are described in Chapters 9 and 15, respectively. Chapter 6 discusses compression tests under hydrostatic pressure and other multi-axial tests. Menges and Knipschild (1982) critically assessed specimen designs in the DIN German standards (www.din.de) for foam mechanical tests. Designs differ between the British (www.bsionline.techindex. co.uk), American (www.astm.org), and German standards, in spite of the intent to generate the same information. ASTM (American Society for Testing and Materials) standards are also published as books, collected according to product areas. Although tensile tests are common for other materials, they are rarely used for foams. This is partly due to the difficulty of gripping foams to apply tensile loads, and partly because few applications involve tensile loads; most foams are weak in tension and fracture easily. In contrast, compressive loading is common, for example in foam cushions and packaging foams loaded by the weight of the contents. Shear occurs if the applied force vector lies in the surface plane of the foam block (Fig. 5.1a). Bending and torsion often occur in foam products. Figure 5.1b shows the geometry of a 3-point bend test. We see later that the stresses in bending or torsion vary with the distance from the neutral surface or axis of twisting of the specimen, so only a small region is under high stress. This contrasts with compression tests, where stresses are intended to be uniform.

F

w F Me

X

d

tal

h

F/2 Foam

F/2

Figure 5.1

Foam

s

F (a)

Moving metal plate

Fixed w metal plate

(b)

Three-point bend test and simple shear test, with dimensions and applied forces shown.

Chapter 5 Simple mechanical tests

87

Bead foam mouldings usually vary in density from the skin to the core, so have inhomogeneous microstructures. Consequently the Young’s modulus and strength vary with position in specimens cut from these mouldings; the interpretation of bend tests on such specimens will be considered towards the end of the chapter. Finally, test methods for high (impact) and low (creep) strain rates are described.

5.2

Stiffness and strength of structures A product that resists loads can be considered as an engineering structure. The engineering concepts of stiffness and strength must be defined and distinguished from their everyday meanings. The compressive stiffness kC of a foam structure is the ratio of the compressive force F to the deflection x of the loading point (while another part of the structure is fixed) kC ⬅

F x

units N m1

(5.1)

Similarly, the bending stiffness kB of a foam beam is the ratio of the bending moment M (unit Nm) to beam curvature C (the inverse of its radius of curvature R, with units m1)

kB ⬅

M MR C

units N m2

(5.2)

The torsional stiffness kT of a foam beam is defined as the ratio of the torque T (Nm) to the angle of twist θ (radians) per length L of beam

kT ⬅

T θ/L

units N m2

(5.3)

Note that the units of stiffness vary depending on the type of loading considered. The strength of a structure is the force or moment or torque needed to cause failure or collapse. Thus the compressive strength of a foam structure is a force F* and the bending strength is a moment M*. These strengths should be distinguished from the tensile strength of a material, with unit Pa (see next section).


5.3

Stress–strain responses and material parameters If mechanical test results are expressed in terms of stress and strain, the results should be independent of the size and shape of the test specimen. However, the shapes of compression (Section 4.1) and shear specimens (Section 4.2) can affect the results. Unless mentioned otherwise, these are engineering stresses, defined in terms of the original dimensions of the foam, and engineering strain, defined for tension as the relative change in length. The true stress in a compression test is the load divided by the current cross-sectional area. However, as foams do not expand much in cross-sectional area on compression, the difference between true and engineering stress is less than for rubbers. Stress–strain data are used for design. Data from a variety of test types is needed to calculate the parameters of foam material models in finite element analysis (FEA). For the latter, it is useful to classify foam stress–strain responses as being one of the following types.

5.3.1 Linearly elastic and isotropic An elastic response means that the stress vs. strain relationship is the same whether the foam is being loaded, kept at a fixed stress, or unloaded. A linear elastic relationship between compressive or tensile stress σx, applied along the x-axis, and the corresponding strain can be described by σx Eεx

(5.4)

where the constant E is the Young’s modulus. In such a tensile or compressive test, there are strains in the y- and z-axis directions, given by εy εz νεx

(5.5)

where ν is Poisson’s ratio. The properties of isotropic foams are the same in all directions, and there are two independent elastic constants. Consequently, if the Young’s modulus and Poisson’s ratio are measured, the shear modulus G can be calculated using G 2E(1 ν)

(5.6)

The linear elastic idealisation is useful for foams at strains less than a few per cent.


89

5.3.2 Elastically non-linear and isotropic Stress–strain relationships, for flexible polyurethane (PU) and ethylene– vinyl acetate (EVA) copolymer closed-cell foams, are usually convex upwards in tension (and in compression, if the compressive stresses and strains are taken to be positive). The non-linearity is partly due to changes in the foam microstructural geometry at high strains. These foams fall into the category of non-linearly elastic materials, with soft biological materials and rubbers. They are sometimes called hyperelastic materials, meaning that the stress–strain relation can be calculated from a strain energy function (see Chapter 6 on FEA).

5.3.3 Anisotropic and elastic Foaming processes often produce elongated cells (Chapters 2 and 3). Therefore, the foam stress–strain response is a function of the direction of the applied stress relative to the foam axes (extrusion direction, rise direction, etc.). A high strain applied to an initially isotropic foam may also cause mechanical anisotropy; the cells may be flattened into a particular plane, so the foam becomes anisotropic. Such materials have more elastic constants than isotropic materials. Chapter 7 considers regular foam models with cubic symmetry, which have three independent elastic constants, for instance a Young’s modulus, a shear modulus, and a Poisson’s ratio. Foams with transverse symmetry (the same properties in any direction perpendicular to the rise direction) have five independent elastic constants, but it is rare for all of these to be determined.

5.3.4 Elastic–plastic If the applied stress exceeds a limit, the yield stress, some of the strain is permanent when the stress is removed. The idealisation works well for metals, but less well for polymer foams, because the slowness of viscoelastic recovery can cause strains to appear permanent. Polystyrene (PS) foams have a compressive yield stress, and a residual compressive strain on unloading. Low-density polyethylene (LDPE) foams, although they show a change in slope in the compressive loading relationship (Fig. 11.12), show no residual strain on unloading. Hardening means an increase in the yield stress with increases in strain. Closed-cell foams harden due to the compression of the cell gas, so the volume strain may be used as the strain measure. The foam tensile strength is the stress at which it fails in tension, with unit Pa. The tensile strength should be independent of the size of the test specimen used.

90 Polymer Foams Handbook 5.3.5 Elastic–brittle When a foam product is bent or twisted, the stress (strain) distribution is non-uniform. In bend tests taken to failure, the flexural strength is the maximum stress in the specimen, calculated assuming the material remains linearly elastic. Figure 5.2 shows the longitudinal stress distribution in a 3-point bend specimen made from a linear-elastic foam with Young’s modulus 10 MPa. The beam cross-section is 50 mm by 50 mm, and the centrally applied force 24 N. Since the beam has a 10 to 1 span to depth ratio, the effects of the 20 mm diameter loading cylinders, in locally disturbing the ideal stress distribution, are small. If the foam fractures at the point A, the flexural strength can be calculated using the elastic stress at that point.

5.3.6 Viscoelastic materials All polymer foams are viscoelastic to some extent. For example, flexible PU foams show hysteresis on unloading (the data falls below the loading data, and energy is absorbed in the loading–unloading cycle). The term viscoelastic implies a combination of a Newtonian viscous response, where the tensile stress σ is proportional to the tensile strain rate dε/dt and the constant η is the tensile viscosity ση

dε dt

(5.7)

and the elastic response of equation (5.4). The response of linear viscoelastic models (see Chapter 19 on creep) can be described using Rigid end roller 20 mm radius

40

80

120 A

0

0

−40

−80 −120 Mirror symmetry at mid-beam

Figure 5.2

FEA prediction of longitudinal stress contours (kPa) in one half of an elastic expanded polystyrene (EPS) beam under 3-point loading (Mills, unpublished).


91

linear differential equations, with mechanical analogues of linear springs and viscous dampers (dashpots). The Voigt mechanical model, often used for creep, is shown symbolically as a spring and dashpot in parallel. The parallel connection implies that the deflection is the same in the spring and the damper. For the mechanical device, the equation between the total force F and the deflection x is F kx n

dx dt

(5.8)

where k is the spring tensile stiffness and n the damper constant. If the mechanical device is hidden in a unit ‘black box’ cube, the force F, applied to a 1 m2 area, can be replaced by the stress σ, and the extension, occurring over a length of 1 m, can be replaced by the strain ε. It is then necessary to rename the two constants as material constants; E is the Young’s modulus and η is a viscosity. Hence the viscoelastic equivalent of equation (5.8) is σ Eε η

dε dt

(5.9)

This constitutive equation is for a linear viscoelastic material, since it only contains the first power of the variables. It would be a non-linear viscoelastic equation if it contained non-integral or higher order powers of the variables. The response of polymers such as PU or LDPE at high strains is non-linearly viscoelastic, as is the response of the foams made from these polymers.

5.3.7 Viscoelastic phenomena Viscoelastic materials exhibit phenomena such as: • Creep, when a constant stress is applied (at time zero) and the strain increases with time. • Stress relaxation, when a constant strain is applied at time zero, and the stress decreases with time. • A phase lag between the stress and strain response, in the so-called dynamic mechanical test. The material is subjected to a small stress that varies sinusoidally with time t σ σ0 sin(ωt δ)

(5.10)

92 Polymer Foams Handbook where ω is the angular frequency and δ is a phase lag angle. The tensile strain varies as ε ε0 sin ωt

(5.11)

When the strain has its maximum value ε0 the stress is σ0 cos δ. In a complete cycle, it is shown by Mills (2005) that Energy dissipated per cycle 2π tan δ Maximum stored ellastic energy

(5.12)

Hence the quantity tan δ is used as a measure of energy dissipated. A graph relating stress and strain is an ellipse; the greater the angle δ, the broader is this ellipse. The extremes are a circle for a purely viscous liquid where δ π/2, and a straight line for an ideal elastic solid where δ 0. The area within the ellipse represents the energy dissipated in unit volume of material in a cycle, which will appear as heat. If the stress of equation (5.10) is divided by the strain of equation (5.11) the result is the dynamic Young’s modulus E*, which is often written as a complex number E* E i E

(5.13)

with real and imaginary parts that are respectively the storage Eand loss E moduli. 5.3.7.1

Strain rate effects

The material properties of polymers change if the strain rate is increased. For example, the compressive yield stress of rigid foams tends to increase linearly with the logarithm of the strain rate in a compression test.

5.3.8 Temperature-dependent properties All polymers have temperature-dependent properties. The effect of increasing temperature on the low strain response of many polymers is equivalent to the effect of increasing the frequency of a sinusoidal input, or increasing the strain rate of a tensile test. The equivalence between temperature and frequency or time can be used in some circumstances to construct viscoelastic master curves, using a method of time–temperature superposition (Ferry, 1961). Dynamic mechanical thermal analysis (DMTA) employs miniature versions of the tests described in this chapter: compression of foam

Chapter 5 Simple mechanical tests 8.0

93

0.20

7.5

0.18

β

0.16

7.0

0.12 0.10

6.0

0.08

5.5

α

0.06

5.0

0.04

4.5

0.02

4.0

tan δ

log E′ (Pa)

0.14 6.5

0.00 −75 −50

−25

0

25

50

75

100

T (°C)

Figure 5.3

DMTA of EVA foam of density 150 kg m3 with a glass to rubber transition at 10°C (Verdejo, 2003).

cubes less than 10 mm side, or 3-point bending tests. Since foams are much softer than solid polymers, the dynamic Young’s modulus (equation 5.12) can be measured in compression, rather than in bending or shear. The temperature can be increased at a constant rate, to find the temperature dependence of the foam elastic moduli. Temperatures, at which tan δ has a maxima, correspond to phase changes in the polymer or the freeing of molecular rotation. At each peak the temperature dependence of the storage modulus is a maximum. Thus the EVA foam in Figure 5.3 has a glass transition at Tg 艑 10°C and a crystalline phase transition at 80°C. At Tg the amorphous phase changes from a high modulus glass to a rubbery material. Transition temperatures can be close to use temperatures. For foams used inside cars, the greenhouse effect of the sloping front and rear windows can cause temperatures up to 130°C. This exceeds the glass transition temperature of PS foam, and can cause the modulus of PU foams to drop by a factor of 50 or more.

5.4

Test types 5.4.1 Uniaxial compressive tests Compressive stress–strain results can depend on a number of factors. 5.4.1.1

Specimen shape and size

The Poisson’s ratio of most foams is approximately zero, meaning there is no lateral expansion of the foam in a compression test.


Rotational symmetry axis

Rigid plate

Rigid plate

Figure 5.4

100

150

50

300

FEA of flexible PVC closed-cell foam, with contours of vertical compressive stress (kPa), at a mean strain of 47% (Mills, unpublished).

Therefore, it is not necessary to consider the effect of friction, at the metal/foam interface, on the stresses in the specimen. However, for closed-cell soft polyvinyl chloride (PVC) foam, the Poisson’s ratio is high, and friction causes a non-uniform pressure distribution across the loaded face. Figure 5.4 shows this for a cylindrical sample of initial height equal to the diameter, with friction coefficient 0.7. The sample sides bulge most at mid-height, and there is a non-uniform stress distribution, with a 300 kPa stress at the location of the original specimen corner. However it is rare to consider the effects of friction on foam tests, and there is no research showing the effect of using low-friction layers on the load interfaces. If the foam specimen was taller, the effects of friction at the ends would be less. However, tall slender foam pillars should not be used for compression tests, because above a critical load they buckle; the deformation mode changes from uniform compression to compression plus bending (a plastic 30 cm ruler buckles if compressed axially). The stress distribution becomes non-uniform, so the nominal stress σ, calculated as F/wd, is no longer uniquely related to the average compressive strain. Compressive tests should be carried out on near-cube specimens, to avoid the risk of buckling. Size effects occur if the specimen dimensions do not exceed 20 times the cell size. There is an exterior layer of weak, cut surface cells on the specimen, of thickness approximately half the mean cell diameter. Neighbouring complete cells may also be affected, and the contribution


95

of these weaker cells on the total stress should be less than 5% of the total. 5.4.1.2

Dynamic effects

If open-cell foams are compressed at impact rates, airflow may contribute to the total stress. This phenomenon is only significant for specimens of side 0.2 m, impacted at a more than 5 m s1 (see Chapter 10 on sport mats). The analysis of the acceleration vs. time trace from a compressive impact rig to extract the foam stress–strain relation is discussed in Section 5.6.1. The effects of sound waves or foam inertia in such tests on rigid or flexible foams are discussed in Section 6.4. 5.4.1.3

Non-uniform strain

When some foams are compressed, a band of highly compressed material forms, surrounded by regions with lower strains. As the compressive deformation increases, the band increases in thickness. This strain inhomogeneity is related to the phenomenon of necking of polymer tensile specimens; a similar analysis can be used to establish the criterion for strain instability. The nominal compressive strain ε, calculated as x/h where x is the displacement and h the initial height, is the average strain in the material. In Chapter 12 on packaging design (Section 3.4), shear bands form at the top of extruded polystyrene (XPS) foam pyramids.

5.4.2 Simple shear tests In the simple shear test (Fig. 5.1a), the upper and lower foam surfaces are bonded to flat metal plates. For the foam specimen to undergo simple shear, its height must remain constant, so the top plate must move parallel to the lower plate. This can be achieved by using a linear bearing to constrain the motion of the top plate. Consequently, as well as the force FT tangential to the plates (Fig. 5.5), there will also be a force FN, normal to the plates. If tensile forces were applied to the ends of unconstrained metal plates via freely rotating joints, the plates would not remain parallel, and the foam would pull the plates closer together. An alternative test arrangement uses two equal foam blocks bonded on either side of a central metal plate. The outer metal plates are constrained to remain parallel at constant separation (Fig. 5.14). The stress distribution in the foam depends on the specimen length to height ratio, the shear strain magnitude, and the material response. A simple analysis will be considered first.

96 Polymer Foams Handbook FN Metal FT

5 10 y

15 20 25

FT

x Metal

2

Figure 5.5

FN

FEA of foam shear with aspect ratio 2:1 at a mean shear strain of 0.25, with shear stress contours (kPa) (Mills, unpublished).

5.4.2.1

Approximate analysis, assuming uniform stress

The approximate analysis assumes that the shear stress is uniform, with value equal to the average shear stress τ– in the specimen. This is related to the force FT, the specimen length L, and its width w (Fig. 5.1b) by τ

FT wL

(5.14)

However, at the free end surfaces of the specimen, there is zero shear stress in the axes of the surface plane. At low shear strains, this plane is aligned with the y-axis, so the shear stress τxy 0. At a moderate shear strain (Fig. 5.5), the end surfaces have rotated, so τxy is no longer zero at the end surface. However τxy is still low near the free surfaces of the foam. As the specimen length L increases, the influence of the ends diminishes, until, for an infinite length specimen, there is no end effect. The shear strain γ is given by γ

X h

(5.15)

where X is the deflection and h the height of the foam block. This is an approximation at high deformation, where the shear strains can become slightly non-uniform (Fig. 5.6). Theory, developed for simple shear flow of incompressible rubbery liquids (Lodge, 1964), relates the principal elongation ratios λa


97

10 20 30 50 60 (a) 0

Rigid plate

5 10 20 Rigid plate (b)

Figure 5.6

FEA predictions of shear stress contours (kPa) in soft foam blocks: (a) aspect ratio 2:1 at shear strain 0.875 and (b) aspect ratio 5:1 at γ 0.525 (Mills, unpublished).

and λb in the foam, in the plane of shearing, to the applied shear strain γ by γ 2 cot 2χ

(5.16)

where λa cot χ

λb tan χ

so λaλb 1

(5.17)

Alternatively, the principal extension ratios are related to γ by

λ1,2 1

γ2 γ2 γ 1 2 4

(5.18)

These equations can be derived by considering a unit circle embedded in the undeformed material. The x, y coordinates of the circle have parametric form x cos θ

y sin θ

(5.19)

In the sheared material the shape becomes an ellipse, with parametric form x cos θ γ sin θ

y sin θ

98 Polymer Foams Handbook The radius rof the ellipse is given by r2 1 2γ sin θ cos θ γ2 sin2 θ

(5.20)

When equation (5.20) is expressed in terms of 2θ, differentiated with respect to 2θ, and equated to zero to find the maximum radius, equations (5.16) and (5.17) result. The angle χ, between the direction of the tensile principal strain and the x-axis, is 45° at low strains, but decreases at higher strains. When γ 1, λa 1.62, and the angle χ 31°. Since open-cell polymer foams are more resistant to tensile than compressive strains, the tensile strains cause significant shear hardening. Typical data for the shear stress and tensile stress in a simple shear test are given in Figure 6.4. 5.4.2.2

Stress distribution at large shear strains

High shear strains cause the shape of the ends of the foam block to change significantly, reducing the stress concentration at the corners. For flexible PU foams, using the hyperelastic model, FEA shows, for a specimen aspect ratio (length divided by its height) of 2:1, the shear stress in its centre was significantly below that given by equation (5.14), and there is tensile deformation in this region. However, for a 5:1 aspect ratio, the shear stress is close to that computed by equation (5.14) in much of the specimen. The steel support plates must be relatively thick to avoid being bent elastically. Therefore the aspect ratio should be at least 5, for the shear test results to be meaningful. There is still a high stress concentration at the lower left and upper right corners of the foam block. Some rigid foams, such as expanded polypropylene (EPP), yield in shear. A yielded region spreads from the ends of the foam block, reducing the stress concentrations at the corners of the specimen. There is no published FEA of such deformation. 5.4.2.3

Shear tests related to sandwich beams

Wada et al. (2003) performed shear tests on foam blocks of varying length to thickness ratios. They used four blocks disposed symmetrically (Fig. 5.7a), to eliminate off-axis loads on the carbon fibre reinforced plastic (CFRP) plates, through which the tensile loads were applied. However, the foam does not undergo pure shear deformation since the outer aluminium plates can move towards the central CFRP plates. Figure 5.7b shows how the apparent shear modulus increases with the core length L to thickness t ratio. The experimental shear moduli reach a plateau when L/t is 4. A theoretical shear modulus of 41.8 MPa was calculated from the measured Young’s


99

Al plate

Force

Foam Foam CFRP plate

CFRP plate

Foam

Force Y

Foam Al plate

Apparent foam shear modulus (MPa)

40

30

20

10 Experiment FEA 0 0

1

2

3

4

5

Block length/thickness

(a)

Figure 5.7

(b)

(a) Specimen design and (b) variation of apparent shear modulus with block aspect ratio (redrawn from Wada et al., 2003 © Elsevier).

modulus and Poisson’s ratio of the acrylic foam; however this estimate may be too high as the foam is slightly anisotropic. Chapter 18 shows that short sandwich beams, subjected to 3-point bending, have a near-constant shear stress in large parts of the core. However, the foam properties affect the stress distribution in the core, and the stress field near the loading points is different. Consequently, this is not a good method to determine foam shear properties. Caprino and Langella (2000) replaced the foam near the three loading points with ‘rigid’ wooden blocks. This complicates matters further, and the results of such tests are only approximate.

5.4.3 Bend tests 5.4.3.1

Introduction

Three-point bend tests can only be used for the more rigid foams. They are rarely used for flexible PU foam, which tends to deform under its own weight, and wraps around the central loading point at high deformation. Bend tests are used to measure the Young’s modulus of metals, since neither an extensometer, nor a correction for the compliance of the load cell is needed. However, given the low Young’s modulus of polymer foams, compression tests can easily generate stress–strain data. Threepoint bend tests are used to determine the tensile strength of EPS bead foams, which are brittle in tension. Tensile tests on EPS require large end tabs, so the pressure from the grips does not cause crushing. A noncontact extensometer is required to measure the tensile strain. Failure in a 3-point bend test initiates in a small, high stress region, so the results are less affected than tensile tests by the random


10mm

Figure 5.8

Failure at bead boundaries in EPS foams under 3-point bending (Moosa and Mills, 1998).

location of large flaws. Bead foam products often fail in bending, so bend tests are used to characterise their strength. As the bending moment increases, the compressive stress may exceed the foam yield stress at locations on the compressive surface, often near the central loading point. Fracture initiates on the tensile surface where the stresses are highest (Fig. 5.8). BS 4370 recommends using a beam of span 300 mm, depth 25 mm and loading point radii of 15 mm. It requires any skins or moulded surfaces to be removed. If any crushing is found, flexural strength calculations should not be made. The following sections consider different approximations to the material behaviour. 5.4.3.2

Elastic stress analysis

If the beam is much longer than it is thick, Euler beam theory applies (Chapter 18 considers the analysis of short beams with stiff skins). If the loading point separation is much greater than the beam depth, the contribution of beam shear to the total deflection can be neglected. Consider a beam that is bent into a circular arc of radius R. The neutral surface is a layer in the beam with length L equal to that in the undeformed beam (L0). Hence the longitudinal tensile strain ex

L L0 L0

(5.21)


101

To centre

R R−y y

z L x

L0 Neutral surface w

Figure 5.9

Geometry of a bent beam, showing a ‘fibre’ of length L, a distance y above the neutral surface.

is zero in the neutral surface. Figure 5.9 shows a fibre in a layer, a distance y above the neutral surface; a comparison of similar triangles gives R/L0 (R y)/L. Hence the tensile strain along the beam, in the direction of the x-axis, is ex

y R

(5.22)

Consequently, ex varies linearly through the thickness of the beam. For beams of symmetrical cross-section, the neutral surface is at the mid-depth, and the maximum and minimum strains, at the top and bottom surfaces, respectively, are given by em

d 2R

(5.23)

where d is the beam depth. If the material is elastic, the longitudinal tensile stress is σx Eex

Ey R

(5.24)

To express this in terms of the applied force, the elastic beam equation EI MR

(5.25)

102 Polymer Foams Handbook is used, where the second moment of area for a beam of rectangular cross-section I wd3/12. If the localised contact stress field of the loading points is ignored, the maximum tensile stress σmax is related to the bending moment M by σmax

6Mmax wd 2

(5.26)

For a 3-point bend test with span S, and force F at the central load point (Fig. 5.1a), the maximum bending moment is at the midpoint Mmax FS/4, so σmax

3FS 2wd 2

(5.27)

If the beam deflection V is small compared with the span, the beam curvature can be approximated by d2V/dx2, where x is the length co-ordinate. Integration leads to an expression for the maximum strain emax

6dVm S2

(5.28)

where Vm is the deflection of the midpoint relative to the end supports. If the angular deflection is large, equation (5.28) must be replaced by one containing elliptic integrals, or FEA used. For the 30 mm deflection limit specified in BS 4370, the 5% strain calculated by equation (5.28) overestimates the true maximum strain by 1.5%; the error is less at lower deflections. The beam surface at the outer loading points is predicted to slide 4.8 mm past the loading cylinders when the deflection is 30 mm, because the beam neutral surface does not change in length, yet the outer ends of the beam rotate. However, if there is foam indentation at the loading points, such sliding will be resisted. 5.4.3.3

Avoiding yield at the loading points

The specimen dimensions and contact roller radius should be chosen to avoid permanent indentation under the loading points. Specimens should be examined after testing; if yielding is noted, the loading radii should be increased or the beam length to depth ratio increased. If yielding occurs, equations (5.27) and (5.28) cannot be used to calculate the maximum stress and strain, respectively. Section 6.3.5 considers the FEA of foam indentation by the side of a metal cylinder. A similar static FEA analysis was performed for cylinder radii from 5 to 20 mm, pressing on a block of 50 mm width

Chapter 5 Simple mechanical tests Mirror symmetry

75

20 Indent or force (N)

103

50

10 mm radius steel cylinder

15 10 5

25

0.25

Foam

0.15 0

0.05 0

0.2

(a)

Figure 5.10

0.4 0.6 0.8 Displacement (mm)

1 (b)

4 mm

(a) Force vs. deflection graphs for cylinder radii (mm) shown, on EPS of density 35 kg m3 and (b) plastic strain contours in the unloaded sample (Mills, unpublished).

and thickness, supported on a rigid table. The EPS foam had density 35 kg m3. Force–deflection graphs, for loading to 1 mm deflection then unloading, are shown in Figure 5.10a. There is non-linearity, indicating some localised yielding, from very low loads. However, to have a significant effect on a bend test, the depth of yielding must be the order of 1 mm. The figure shows that a larger cylinder radius allows larger loads before a 1 mm deflection. The residual deflections, after unloading, decrease slightly as the cylinder radius increases. Figure 5.10b shows the predicted residual plastic strain distribution in the unloaded specimen, for the 10 mm radius cylinder. These strains are directly under the cylinder, but non-uniform. 5.4.3.4

Large-scale yielding

Rigid foams such as EPS tend to yield at a lower stress in compression than in tension. Consequently, in a bending test, yielding commences on the compressive side. The longitudinal strain e in the beam is still given by equation (5.22). In the elastic part of the beam the longitudinal stress σ is given by equation (5.23). In the analysis, the foam compressive yield stress is assumed to be constant (C). Figure 5.11 shows the variation of the longitudinal stress with distance through the beam. The condition, of no net force parallel to the length of the beam, means that the tensile area of the graph equals the compressive area, so CL

1 (T 2

C)(d L)

where L is the depth of yielding from the compressive surface.

(5.30)


T

Stress

Area = tensile force

0

L Area = compressive force

d

N

y

C

Figure 5.11

Assumed variation in longitudinal stress across a partly yielded beam (Moosa and Mills, 1998).

The peak tensile elastic stress T can be expressed in terms of L ⎛ d L ⎞⎟ ⎟ T C ⎜⎜ ⎜⎝ d L ⎟⎟⎠

(5.31)

The distance N of the neutral surface from the yielded surface is N

Cd LT C T

(5.32)

The moment M of the stress distribution is ⎛ M L⎞ C T CL ⎜⎜ N ⎟⎟⎟ (N L)2 (d N)2 ⎜ ⎟ w 2⎠ 3 3 ⎝

(5.33)

Non-linearity in the beam load–deflection relation can therefore indicate local yielding, rather than viscoelasticity. If the foam compressive yield stress is known, the maximum tensile stress T can be calculated from the moment M at the centre of the beam. If plasticity commences at a moment M0, for smaller moments the beam radius of curvature R is given by equation (5.25), so M E T I R dN

(5.34)


105

5.4.4 Torsion tests Torsion tests are rarely performed, due to the lack of torsion attachments on test machines. Nevertheless, torsion tests on thin-walled cylinders allow the measurement of shear response without having an end effect.

5.5

Testing products with a density gradient Chapter 18 deals with sandwich panels, which have stiff skins separated by a foam core. The density variations through EPS mouldings (Fig. 4.5) produce a related type of structure, which means that mechanical tests on such mouldings must be interpreted carefully.

5.5.1 Tensile or compression tests on EPS A foam moulding with length and width thickness, having a density variation from the surface to the core, can be treated as a number of uniform layers. Tensile tests require long specimens, so the tensile axis is likely to be in the plane of the moulding. Consequently the same extension or strain is applied to the layers of the specimen. The tensile force, applied to a unit width of layer, is the product of its Young’s modulus Ei, its thickness ti and the strain. The total tensile force, divided by the total specimen thickness, is the applied tensile stress. The tensile modulus ET is this stress, divided by the (constant) tensile strain, so

ET

∑ Ei ti i

∑ ti

(5.35)

i

This is the same as the upper bound for the modulus of a composite material containing volume fractions Vi of material i Emax

∑ EiVi

(5.36)

i

Compressive tests are likely to be carried out with compression axis perpendicular to the moulding surfaces. Consequently, the layers of the mouldings are loaded in series, with the same stress acting on each

106 Polymer Foams Handbook layer. The compressive deflections of each layer are added to give the total deflection, so the compressive Young’s modulus EC is given by t

1 EC

∑ Ei

i

i

∑ ti

(5.37)

i

This is the same as the lower bound for the modulus of a composite material. Using density data for an EPS tile (Fig. 4.5) equations (5.35) and (5.37) give EC 35.5 MPa, and ET 40.0 MPa. Hence the inhomogeneous product acts as a material with anisotropic elasticity.

5.5.2 Bend tests on EPS Bend tests require long specimens, so their length axis lies in the plane of the EPS moulding. It is assumed that the foam remains linearly elastic. For a beam radius of curvature R, the longitudinal stress σ in a layer with co-ordinate y, measured from the lower surface of the beam, is σ

E(y) (y N) R

(5.38)

where N is the height of the neutral surface above the lower beam surface. As Poisson’s ratio is approximately zero (Section 5.6.5), there is no lateral curvature of the top surface of the bent beam. The value of N is determined by the condition that there is no net longitudinal force y max

F

∫

σ w dy 0

(5.39)

y min

On substituting equation (5.38) this gives y max

∫

E(y)(y N) dy 0

(5.40)

y min

The density variation with y is measured, and the Young’s modulus profile (Fig. 5.12a) computed using equation (11.36). The position

50

Distance from top surface y (mm)

Distance from top surface y (mm)


40 30

Neutral surface

20 10 0 20

(a)

30 40 50 60 Young’s modulus (MPa)

Figure 5.12

70

107

50 40 30 20 10

0 2 1.5 1 0.5 0 0.5 (b) Stress (MPa)

1

1.5

2

(a) Computed Young’s modulus variation through a 50 mm thick EPS tile of density 80 kg m3, and the neutral surface position in a beam cut from the tile and (b) stress distribution in the beam, when close to fracture (Moosa and Mills, 1998).

of the neutral surface is found by a numerical search. The external bending moment M is related to the distribution of stresses on the beam section by y max

MR

∫

E(y)(y N)2 dy

(5.41)

y min

When R is eliminated between equations (5.34) and (5.41), the stress is given in terms of the applied moment M. Figure 5.12b shows the stress distribution through the beam. Due to its complex shape, a calculation of the foam flexural strength using equation (5.27) is likely to be in error.

5.6

Test equipment ‘Universal’ testing machines are commonly used to carry out tensile, bending, and shear tests. As foams are time dependent, the test conditions should be appropriate to the application. If a packaged product is stored, some protective foam experiences a constant load (creep). If the package is dropped, some foam is impact loaded. It is rare that strain is applied at a constant rate, yet, as most test machines have a constant crosshead velocity, this is an easy test to perform.

108 Polymer Foams Handbook 5.6.1 Compressive impact Commercial equipment for determining cushion curves is only suitable for large blocks of foam. It is often preferable to construct equipment, with appropriate striker mass and instrumentation sensitivity, for compressive impact tests on small foam specimens (Fig. 5.13). In most falling-mass impact testing machines, the striker acceleration is monitored. To minimise resonance, the striker is a compact cylinder of cast iron, which has a higher internal friction than steel. The striker carriage is made from aluminium honeycomb with bonded aluminium skins; it is stiff in bending, yet has mass of only 0.25 kg. Consequently, 95% of the impact mass is in the striker. Mills (1994) used a Kistler ‘Piezotron’ accelerometer with integral electronics, attached to the striker. The cable for a voltage output is simpler than for a charge output. An analogue-to-digital (A to D) converter digitises the signal at 12 bit accuracy, typically every 1.0 µs. Electrical filtering of the signal is not recommended, since it can hide instrumentation problems, such as ringing from a loose accelerometer mount. If a quartz crystal load cell is used below the foam, the support table mass must be kept low, so that the resonant frequency of the system

Laser extensometer

Accelerometer

5 kg mass

Aluminium honeycomb

PTFE bearings

Steel plate

Foam

Guide cables

Steel plate Quartz load cell Large steel plate

Figure 5.13

Instrumentation for compressive impact (updated from Mills, 1994).


109

remain above 10 kHz. A non-contact displacement detectors, based on laser triangulation, has a range of 50 mm and sampling rate of 10 kHz. This was used by Verdejo and Mills (2004) to monitor the foam impact. The impact velocity can be measured just prior to impact using photocells and a digital timer. The striker guidance system should have low friction (PTFE (polytetra fluoro ethylene) bushes loosely fitting on guide cables), so the measured velocity is at least 98% of the theoretical velocity. Two, parallel, guidance cables are easy to align. As they have a moderate horizontal stiffness, if the centre of mass of the striker is not directly above the centre of the foam specimen, the lateral forces on the cables during the impact are moderate. Consequently the frictional forces remain low. Commercial systems, with the striker guided by cylindrical linear bearings running on four vertical steel rods, are difficult to align perfectly; the rods are stiff horizontally, so high frictional forces arise if specimen alignment is not perfect. The acceleration signal baseline is set to zero when the striker is falling freely. A rise in the acceleration signal by a few units indicates the start of the impact; the impact time t, and the foam deflection are set to zero. It is difficult to test ultra-soft foams, since the very low acceleration signal during initial contact may not be detected. To capture the whole trace, the acceleration range must be high, as the peak acceleration when the foam bottoms out will be high. If the accelerometer output exceeds the range of the A to D converter, the signal will be clipped. For testing soft foams, the falling mass must be reduced, the foam area increased, and a high resolution A to D converter used. Figure 5.14 shows a schematic variation of the striker acceleration a with time. The numerical integration of the acceleration uses the trapezium rule Vnew Vold 0.5(aold anew)∆t

(5.42)

and the known initial velocity Vi at the start of impact. The position of the foam upper surface x is xnew xold 0.5(Vold Vnew)∆t

(5.43)

with the condition that x 0 when t 0. The compressive strain ε in the foam is calculated as x/h. Newton’s second law can be used to calculate the force on the foam. The foam mass, usually less than 10 g, is negligible compared with the striker mass m ⬇ 5 kg, so the force needed to accelerate the foam is neglected. Hence the compressive engineering stress is σ

ma A

(5.44)

110 Polymer Foams Handbook 250 200 Acceleration (m s−2)

150 100 50 0

Velocity (0.01 m s−1)

−50 −100 −150

Figure 5.14

Position (0.1 mm) 0

5

10 Time (ms)

15

20

Schematic of integration of the striker acceleration to produce its velocity and position.

where A is the top surface area of the foam and a is the striker acceleration. Calibration checks can be made by impacting a truncated cone of Plasticine (modelling clay), which has a very low coefficient of restitution. The computed velocity at the end of the acceleration trace should be very low. The maximum displacement should be equal to the change in the height of the cone.

5.6.2 Tensile or shear impact In some tensile or shear impact test rigs, a falling weight is caught by part of a lower clamp, attached to the specimen. Figure 5.15a shows the equipment for simple shear tests on flexible PU foams. A 5 mm thick layer of LDPE foam is used on the catcher plate to prevent metalto-metal impact causing ‘ringing’ in the metal structures. The two foam specimens are constrained to deform by simple shear. Their length to height ratio of 6:1 means the majority of the foam is under a uniform shear stress (Section 4.2). Results for PU rebonded foam (Fig. 5.15b) show a non-linear response, with some hysteresis. Tensile impacts were used for fracture toughness measurements (Chapter 15).

5.6.3 Creep For a foam to undergo uniform compressive strain, rather than buckle, its initial shape must approximate a cube, while the upper and lower


111

Foam Foam Drop mass

To force cell

Metal

V Catcher plate (a) 100

315

Shear stress (kPa)

80 273 60 196 40 115 20

0

0

0.05

0.1

0.15 0.2 Shear strain

0.25

0.3

0.35

(b)

Figure 5.15

Shear impact: (a) rig and (b) results for chip foam of density 225 kg m3 for drops (mm) (Mills and Lyn, 2004).

surfaces must remain parallel. In a rig developed by Mills and Gilchrist (1997a) a linear bearing constrains the moving top anvil to remain parallel with the fixed lower anvil (Fig. 5.16). The foam thickness is monitored with a Solatron linear encoder, which uses a ruled grating on a silica rod. Its output at a fixed displacement is constant, whereas the output of an LVDT (linear variable-displacement transducer) drifts slightly if the excitation voltage changes. The creep stations are kept in a temperature-controlled box. For a 25 mm cube specimen size, direct masses of up to 5 kg were suspended from the upper (moving) plate via a lightweight cage. The 2 N force exerted by the mass of transducer core and the loading cage was counterbalanced to 0.1 N to allow creep stresses down to 1 kPa to be applied, and to allow recovery measurements when the creep load is removed.

112 Polymer Foams Handbook Pulley Clamp

Thread Counterbalance weight Linear encoder

Linear bearing Aluminium angle section

Foam

Picture frame stirrup

Weight

Figure 5.16

Compressive creep rig (updated from Mills and Gilchrist, 1997a).

5.6.4 Compression set The shoe industry uses the foam compression set as one selection criterion. The test standard EN ISO 1856 (1996) requires a constant compressive strain of 50% to be maintained for 22 h, at 23°C or 70°C, then the residual strain measured 30 min after releasing the sample. The test pieces are 50 mm wide and long, and 25 mm thick. In EVA foams, cell air is lost in the 22 h period when the foam gas pressure exceeds atmosphere pressure. There is incomplete reversal of this process after 30 min. The gas loss is affected by the size of the block tested, so the compression set value depends on the block size.

5.6.5 Poisson’s ratio It is difficult to make Poisson’s ratio measurements during impact, except by photographic means. However it is relatively easy to make measurements at slower rates, using a constant strain-rate machine. In a uniaxial compression test (Fig. 5.17), the spring-loaded cores of LVDT transducers exert a force of about 1 N. The 7 mm by 15 mm flat PTFE ends exert a pressure of about 10 kPa on the foam, well below the compressive collapse stress of most foams. Tests were limited to 40% compressive strain since the transducer position is fixed


113

Moving cross-head Load cell Spring-loaded core

PTFE plate Foam

LVDT

LVDT Metal base plate

Figure 5.17

Instrumentation for Poisson’s ratio measurement on a compression testing machine (redrawn from Mills and Gilchrist, 1997b).

relative to the lower platen. A vertical LVDT transducer, of 50 mm range, monitors the position of the test machine cross-head. The data was sampled with a Pico 14 bit A to D converter, made at intervals of typically 0.1 s, and stored on a microcomputer. The Poisson’s ratio of EPS foam of density 50 kg m3 is initially about 0.08. When yielding occurred at a strain of 5%, the lateral strain became constant 0.2%, indicating a change in deformation mechanism. The low-strain Poisson’s ratios for XPS foams of densities 30 and 44 kg m3 were 0.10 and 0.13, respectively, showing that the lack of voids at bead boundaries only slightly increased Poisson’s ratio. In contrast, the Poisson’s ratio of a 24 kg m3 density LDPE foam, compressed on a 1 min timescale, was 0.17 (Mills and Gilchrist, 1997b).

5.6.6 Humidity and temperature control It is important to control the temperature in any type of polymer testing (Turner, 1983), because of the temperature sensitivity of the results. Humidity can quickly affect the properties of open-cell PU foams because of the high surface area and low diffusion distances. Experiments, in which the mass of PU foam is monitored in a humidity cabinet, indicate the kinetics of the process. Chapter 20 considers water absorption further.

References BS 4370: Part 4 (1991) Methods of Tests for Rigid Cellular Materials, British Standards Institution, London.

114 Polymer Foams Handbook Caprino G. & Langella A. (2000) Study of a three-point bending specimen for shear characterisation of sandwich cores, J. Comp. Mater. 34, 791–814. Ferry J.D. (1961) Viscoelastic Properties of Polymers, Wiley, New York. Lodge A.S. (1964) Elastic Liquids, Academic Press, London. Menges G. & Knipschild F. (1982) Stiffness and strength-rigid plastics foams, Chapter 2, in Mechanics of Cellular Plastics, Ed. Hilyard N.C., Applied Science. Barking, England. Mills N.J. (2005) Plastics: Microstructure and Engineering Applications, 3rd Ed. Butterworth Heinemann, London. Mills N.J. (1994) Impact response, in Low Density Cellular Plastics, Eds. Hilyard N.C. & Cunningham A., Chapman and Hall, London. pp. 270–318. Mills N.J. & Gilchrist A. (1997a) Creep and recovery of polyolefin foams – deformation mechanisms, J. Cell. Plast. 33, 264–292. Mills N.J. & Gilchrist A. (1997b) The effects of heat transfer and Poisson’s ratio on the compressive response of closed-cell polymer foams, Cell. Polym. 16, 87–119. Mills N.J. & Lyn I. (2004) Design and performance of Judo mats, in The Engineering of Sport 5, Vol. 2, Eds. Hubbard M., Mehta R.D. & Pallis J.M., ISEA, Sheffield, pp. 495–502. Moosa A. & Mills N.J. (1998) Analysis of bend tests on polystyrene bead foams, Polym. Test. 17, 357–378. Turner S. (1983) Mechanical Testing of Plastics, Godwin, Harlow, UK. Verdejo R. & Mills N.J. (2004) Simulating the effect of long distance running on shoe midsole foam, Polym. Test. 23, 567–574. Verdejo R. (2003) Gas loss and durability of EVA foams used in running shoes, Ph.D. thesis, University of Birmingham. Wada A., Kawasaki T. et al. (2003) A method to measure shearing modulus of the foamed core for sandwich plates, Compos. Struct. 60, 385–390.

Chapter 6

Finite element modelling of foam deformation

Chapter contents 6.1 Introduction 6.2 Elastic foams 6.3 Crushable foams 6.4 Dynamic FEA (explicit) Summary References

116 117 127 140 143 144


6.1

Introduction 6.1.1 FEA packages Finite element analysis (FEA) is used to find the stress distribution for complex geometries. This chapter explores the background to foam material models in FEA; other aspects of FEA are covered in texts such as Shames and Dym (1985). Experiments to validate the models will be critically examined. Further examples of use of FEA occur in the case studies (Chapters 9, 13, 14, 16, and 21) and the analysis of foam indentation (Chapter 15). The choice of FEA package may be determined by cost. ABAQUS, widely available in universities, provides detailed explanations of the foam models. Updates, usually on an annual basis, have changed the foam models; consequently the pre-2002 modelling of crushable foams is largely ignored here. Other FEA packages, such as LS-DYNA or RADIOSS, offer a wide range of foam material models, but give little information on their origin or internal working. The automotive industry uses FEA for the design of car bodies, and the modelling of occupant protection with rigid foam padded components. It is an advantage if the same FEA programme can model the deformation of the steel structure, the rigid foam padding, and the occupant kinematics and injury criteria. Hence they not tend to use ABAQUS.

6.1.2 Static vs. dynamic FEA For static analysis, the implicit method iterates the solution until it converges, at each deflection step of the problem. It relates forces to deflections, but does not consider the acceleration of masses. For dynamic problems, the explicit method estimates forces, then applies them to the masses to calculate the accelerations in the next time step. Explicit FEA is needed to model the vibration of foam components or sound waves in impacts. Although explicit FEA appears to be the natural method for impact problems, Masso-Moreu and Mills (2003) used the implicit method to analyse impacts on foam packaging because: • The foam mass was insignificant compared with the product mass. Consequently the foam acts as a non-linear, massless spring, and the product acceleration can be computed later from the force on the foam. • Foams such as polystyrene (PS) are nearly strain-rate independent. • In experimental data, there is little sign of elastic stress waves in the foam. • The explicit method can predict large force oscillations if contact conditions are not carefully selected, whereas the implicit method tends to predict smooth force vs. deflection relationships.

Chapter 6 Finite element modelling of foam deformation

117

With the introduction of generalised contact conditions, and other changes, in ABAQUS version 6.5, explicit analysis has improved. Consequently it is now preferable.

6.1.3 FEA material models FEA material models are well established for metals or rubbers, where the constitutive equations for deformation have been researched exhaustively. Some of these equations have been adapted for foams, without being fully validated. The two main types of foam material models are: 1. Non-linear elastic: where the response is calculated from a strain energy function. Such material models are called hyperelastic (Ogden, 1997) to contrast with linear-elastic models. Alternative models curve-fit particular types of stress–strain response. 2. Elastic–plastic: where compressive yielding causes permanent foam densification (crushing). These models are for isotropic materials. However, many polyurethane (PU) and polyvinyl chloride (PVC) foams are slightly anisotropic due to the foam rise during production. Extruded polystyrene (XPS) foam for building applications is deliberately made anisotropic to maximise the through-sheet compressive strength. As FEA models for anisotropic foams do not exist, the modelling of such foams is approximate. The extra factor of viscoelasticity will be considered in Chapter 19. Although foams have complex microstructures on the cell size scale (typically 0.5 mm), they are homogeneous on a larger scale. FEA treats the material as a continuum. It only calculates the forces between elements at the mesh points, and the mesh size is usually larger than the foam cell size. The mesh size used is a compromise between accuracy, requiring a fine mesh, and a rapid solution, requiring a coarse mesh. The product dimensions, material moduli, and densities are input to FEA as numbers, rather than physical units. Consequently, a consistent set of units must be used. SI units are preferred, so the Young’s modulus of an expanded polystyrene (EPS) is entered as 1.0 ⫻ 107, with implied units of Pa. In dynamic FEA, units for acceleration and mass must also be considered. The large deformation option must be used, as changes in foam geometry alter the influence of the external forces.

6.2

Elastic foams 6.2.1 Curve fitting vs. strain energy functions Rusch’s (1970) shape function is a typical empirical curve-fit of uniaxial compression stress σ vs. strain ε data, using

118 Polymer Foams Handbook F(ε) ≡

σ ⫽ aε p ⫹ bεq Eε

(6.1)

where E is the Young’s modulus and a, b, p, q are further constants. These five independent constants are sufficient to make a reasonable fit to most data. The curve-fitting parameters are not linked to deformation mechanisms, so cannot be compared with foam microstructural variables. It is not clear how some foam material models in LS-DYNA, which use curve fitting, predict the foam response under complex strain states. The large strain response of flexible foams cannot be described by linear-elastic models; a compressive stress equal to Young’s modulus would cause the height of a block of such a material to decrease to zero; the resulting infinite density is physically impossible. Hence, the stress–strain relationship for uniaxial deformation must be nonlinear, with a slope that increases with increasing strain magnitude. Such foams can be approximated as hyperelastic solids, if their timedependent mechanical properties and hysteresis are ignored. However, efforts to uncover the elastic response underlying the large strain compression of PU foams (Schrodt et al., 2005), by repeatedly cycling the foam and waiting stress relaxation to equilibrium, failed. Consequently, any elastic model is an approximation. Once the parameters of the strain energy function U are determined, the stress components are obtained by partial differentiating U with respect to the principal extension ratios λi σi ⫽

∂U ∂λi

where the suffix i runs from 1 to 3. If a sphere of unit radius is inscribed in a solid, then the solid uniformly deformed, the sphere becomes an ellipsoid, with axes λi. For non-uniform deformation, the initial sphere is infinitesimal in size. λi are the principal extension ratios and there are no shear strains in the principal axes. Note that Rusch’s shape function is not a strain energy function. Micromechanics models (Chapters 7 and 11) can predict foam stress–strain responses, but they do not lead directly to a particular form of strain energy function (eventually this may be possible). However, the parameters of a strain energy function can be adjusted to fit the stress–strain predictions of the model.

6.2.2 Strain energy function for rubbers The strain energy functions for rubbers have influenced those used for elastic foams. Rubbers are effectively incompressible in most


119

situations, so Poisson’s ratio can be taken as 0.5. The bulk modulus, determined by the strength of the van der Waals forces between the polymer chains, is about 2 GPa. The shear modulus is determined by the crosslink density, so ranges from 0.1 to 5 MPa. Rivlin (1948) proposed a strain energy function U, expressed in terms of invariant functions I of the extension ratios. Invariant means that the function does not change with the choice of axes. He selected three invariant functions I that are integral powers of λi I1 ≡λ12 ⫹ λ22 ⫹ λ32

I2 ≡ λ12λ22 ⫹ λ22λ32 ⫹ λ32λ12

I3 ≡ λ12λ22λ32

As I3 ⫽ 1 for incompressible materials, U is a function of I1 and I2. For uniaxial extension in the x direction to an extension ratio λ, the true stress σx is computed from U by differentiation ⎡ ∂U ∂U ⎤⎥ σx ⫽ (λ 2 ⫺ λ⫺1) ⎢⎢ ⫹ λ⫺1 ∂I2 ⎥⎥⎦ ⎢⎣ ∂I1

(6.2)

For typical rubbers, the partial differentials in equation (6.2) are nearly constant, so are given the symbols C1 and C2, respectively. C1 is related to the density of network chains per unit volume, whereas C2 is related to the density of dangling chains (only connected to the network at one end). Consequently the tensile stress–strain relation has a simple form, called the Mooney–Rivlin equation σx ⫽ C1(λ 2 ⫺ λ⫺1) ⫹ C2 (λ ⫺ λ⫺2 )

(6.3)

6.2.3 Ogden strain energy function for elastic foams Ogden (1972) proposed a strain energy function, containing nonintegral powers of the strain invariants, for slightly compressible rubbers. It has been subsequently adapted for highly compressible low-density foams. In the ABAQUS Users Manual (2004), the function is written as

U ⫽

⎤ 2µi ⎡⎢ αi 1 ⫺αi βi α α (λ1 ⫹ λ2 i ⫹ λ3 i ⫺ 3) ⫹ (J ⫺ 1) ⎥⎥ ⎢ 2 βi ⎥⎦ i ⫽1 αi ⎢⎣ N

∑

(6.4)

In this complex equation, λI are the principal extension ratios, the — measure of the relative volume J ⫽ λ1 λ2 λ3 ⫽ 兹 I3 , the µi are shear

120 Polymer Foams Handbook moduli, N is an integer, while the exponents αi and βi need not be integral. The initial shear modulus is given by

∑

µ⫽

i ⫽1, N

µi

(6.5)

The Mooney–Rivlin equation (6.3) for rubber is a special case of the Ogden strain energy function with N ⫽ 2, α1 ⫽ 2, α2 ⫽ ⫺2, and β ⫽ ⬁. The second bracketed term in equation (6.4) is for volume strains. The coefficients βi that determine the compressibility are related to Poisson’s ratio νi by βi ⫽

νi 1 ⫺ 2νi

(6.6)

If Poisson’s ratio is zero, so are the βi. The initial bulk modulus κ is given by κ⫽

⎛1 ⎞ 2µi ⎜⎜ ⫹ βi ⎟⎟⎟ ⎜⎝ 3 ⎟⎠ i ⫽1, N

∑

(6.7)

The parameters cannot be predicted from the polymer properties, the foam density, and the deformation mechanisms. Instead they must be calculated by curve-fitting experimental data of the types described in the following sections.

6.2.3.1

Uniaxial response

For uniaxial tension or compression along the first axis, the applied stress is σ1 ⫽

∂U 2 ⫽ ∂λ1 λ1

µi αi (λ1 ⫺ J⫺αi βi ) α i ⫽1, n i

∑

(6.8)

For open-cell foams under uniaxial compression, it is a reasonable approximation that Poisson’s ratio ν ⫽ 0, so β ⫽ 0. Hence, the stress is given by σ1 ⫽

2 λ1

µi αi (λ1 ⫺ 1) i ⫽1, n αi

∑

(6.9)


121

15 Tension 10 Kelvin

Stress (kPa)

5

0 −5

8

Compression

−2

−10

2 20

−15 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Strain λ−1

Figure 6.1

Uniaxial compression and tension predictions of N ⫽ 1 Ogden model, for µ ⫽ 10 kPa, α ⫽ ⫺2, 2, and 8, and for µ ⫽ 40 kPa, α ⫽ 20, compared with the Kelvin model predictions for R ⫽ 0.025 and E ⫽ 50 MPa (Mills and Gilchrist, 2000b).

If α ⫽ 2, equation (6.9) predicts an almost-linear tensile response, but non-linear compressive response (Fig. 6.1). When α ⫽ 8, a stress upturn occurs at low tensile strains, but in compression at strains ⬍⫺0.5. If α ⫽ ⫺2, there is a near plateau stress in tension, and a non-linear response at low strains in compression. 6.2.3.2

Simple shear response

For simple shear, where there is no volume change, the shear stress τ and normal stress T (see Fig. 5.5) are related to the shear strain γ by

τ ⫽

T ⫽

∂U ⫽ ∂ε

⎪⎧⎪ 2γ ⎨ 2 ⎪ 2(λ ⫺ 1) ⫺ γ 2 j⫽1, 2 ⎪ ⎪⎩ j

⎪⎫⎪ µi αi (λj ⫺ 1)⎬ ⎪⎪ i ⫽1, n αi ⎪⎭

(6.10)

⎪⎧⎪ ⎪⎫⎪ 2(λj2 ⫺ 1) µi αi (λj ⫺ 1)⎬ ⎨ 4 ∑ ⎪ 2λj ⫺ λj2 (γ 2 ⫹ 2) i ⫽1, n αi ⎪⎪ j ⫽1, 2 ⎪ ⎪⎩ ⎪⎭

(6.11)

∂U ⫽ ∂γ

∑

∑

∑

where the principal extension ratios λj in the plane of shearing are related to the shear strain γ by


λ1, 2 ⫽ 1 ⫹

γ2 γ2 ⫾ γ 1⫹ 2 4

(6.12)

The normal stress contributions in equation (6.11), for j ⫽ 1 and 2, are odd functions of strain. The normal stress is a positive, even function of strain when α ⬎ 2, it is zero when α ⫽ 2, and it is a negative, even function of strain when α ⬍ 2. 6.2.3.3

Hydrostatic compression response

The pressure p is related to the isotropic extension ratio λ ⫽ λ1 ⫽ λ2 ⫽ λ3 by

p⫽

6.2.3.4

2 λ3

µi α (λ i ⫺ λ⫺3αi βi ) α i ⫽1, n i

∑

(6.13)

Fitting data for uniaxial compression

Most authors use the Ogden function with N ⫽ 2 and positive shear moduli to fit data for flexible foams. Better curve fitting can be achieved by using N ⬎ 2, but the predicted stress–strain response can deviate wildly from the expected behaviour when it is extrapolated outside the data range, particularly if negative shear moduli are used. For example, Lemmon et al. (1997) used N ⫽ 3 and two negative shear moduli to describe the uniaxial compression response of ethylene–vinyl acetate copolymer shoe foam; the predicted stress became tensile at high compressive strains! No physically realistic deformation mechanism would produce a negative shear modulus. Setyabudhy et al. (1997) fitted uniaxial compression data for a PU seating foam of density 41 kg m⫺3, with α1 ⫽ 17.4, µ1 ⫽ 18.3 kPa, α2 ⫽ ⫺2.0, and µ2 ⫽ 0.21 kPa. The µ2 term has a significant effect on the predicted stress at high compressive strains. Mills and Gilchrist (2000b) fitted a range of shear, compression, and intermediate tests on an open-cell polyether PU slabstock foam of density 38 kg m⫺3 (abbreviated as PU38) using similar values: α1 ⫽ 20, µ1 ⫽ 20 kPa, α2 ⫽ ⫺2, µ2 ⫽ 0.2 kPa, and ν ⫽ 0. Uniaxial compressive data measured along the rise direction at 24°C at a compressive strain rate of 3.3 ⫻ 10⫺3 s⫺1 is shown in Figure 6.2. There is considerable hysteresis between the loading and unloading response, and the ‘plateau’ stress, for compressive strains ⬎10%, cannot be fully matched by the model. Lyn and Mills (2001) used values of α1 ⫽ 8 and α2 ⫽ ⫺2 for a rebonded PU foam of density 57 kg m⫺3, which


123

20

15 Stress (kPa)

N=2 N=1 10

5

0

Figure 6.2

0

0.1

0.2

0.3 0.4 0.5 Compressive strain

0.6

0.7

0.8

Uniaxial compression stress–strain data (thin curves) for PU38 foam at 24°C, for a strain rate of 3.3 ⫻ 10⫺3 s⫺1 compared with Ogden function predictions for N ⫽ 1, µ ⫽ 10 kPa, α ⫽ 8, and for N ⫽ 2, α1 ⫽ 20, µ1 ⫽ 20 kPa, α2 ⫽ ⫺2, and µ2 ⫽ 0.2 kPa (Mills and Gilchrist, 2000b).

has a more linear compressive response than the PU38 foam. Hence, the α values for PU foam is a function of the microstructure. The approximation that Poisson’s ratio is zero is widely used. For low-density PU foams it is 0.1–0.2 for small compressive strains, then the lateral strain becomes constant at higher strains. However, it is close to 0.5 in tension (Mills and Gilchrist, 2000a). 6.2.3.5

Fitting data for shear and compression tests

The ABAQUS Problem Manual (1998) showed how to fit a particular set of ‘experimental’ compressive and simple shear data (source unknown) with the hyperfoam model, using the parameters α1 ⫽ 8.88, α2 ⫽ ⫺4.82, µ1 ⫽ 1.65, and µ2 ⫽ 0.0023. The data presumably is for a PU foam, with modulus units likely to be kg cm⫺2; that is, µ1 ⫽ 16.5 kPa. There are significant deviations in fitting the shear stress data, while the fit of the uniaxial compressive data is good. Figure 6.3 shows the test rig used to subject a foam specimen, with length/height L/H ⫽ 2, to simple shear, or simultaneous compression and shear (Chapter 5 explains that L/H should be larger to avoid end effects). For simple shear deformation, Figure 6.4a shows the variation of the tensile and shear stresses for PU38 foam. The


Compressive and shear test rig

X- and Y- axis stepper motors

Foam Rigid support

Mounted on two linear bearings so cannot rotate Compressive 50 N load cells

Test rig for measuring shear and tensile forces in foam subjected to shear plus compression (Mills and Gilchrist, 2000b).

20

20

15

15

10

Shear Tensile

5 0

Stress (kPa)

Stress (kPa)

Figure 6.3

0

0.1

(a)

Figure 6.4

0.2

0.3

Shear strain

0.4

10

(b)

Tensile N = 2

Shear N = 1

5 0

0.5

Shear N = 2

Tensile N = 1

0

0.1

0.2

0.3

0.4

0.5

Shear strain

(a) Shear and tensile stress data for simple shear loading and unloading PU38 foam and (b) predictions for the Ogden models of Figure 6.2 (Mills and Gilchrist, 2000b).

predictions are close to the experimental data for strains ⬍0.25, but too high at greater strains (Fig. 6.4b). The α1 ⫽ 20 term dominates the tensile non-linear response, mainly due to cell edge stretching, while the α2 ⫽ ⫺2 term dominates the compressive non-linear response, due to edges touching at high strains. For the three experiments, there is a better fit for unloading rather than loading data.


125

CL

(a)

Figure 6.5

(b)

Deformation fields for plane-strain indentation: (a) predictions for α ⫽ 8, µ ⫽ 10 kPa, ν ⫽ 0, λ ⫽ 0.75, at ε ⫽ ⫺0.6 and (b) data for PU38 foam at ε ⫽ ⫺0.6 (CL is the sample centre line) (Davies et al., 2000).

The Ogden function with N ⫽ 1, α ⫽ 8, and µ ⫽ 10 kPa, which represents the uniaxial tensile and compressive response of the Kelvin foam micromechanics model for deformation in the [100] direction (Fig. 6.1), does not fit the data for PU38 foam as well as the N ⫽ 2 model.

6.2.4 Validation of FEA: plane-strain indentation of flexible foams The plane-strain indentation of a flexible PU foam by a cube (Mills and Gilchrist, 2000b) was used to validate a hyperelastic model. The foam block had a width:depth:height ratio of 3:1:1 and the side of the indenting cube equalled the foam height. The front foam surface remains flat, because the compressive Poisson’s ratio of the foam is zero. Cube-shaped plane-strain elements were used in ABAQUS 5.8 standard, with hyperfoam material parameters α1 ⫽ 20, µ1 ⫽ 20 kPa, α2 ⫽ ⫺2, µ2 ⫽ 0.2 kPa, and ν ⫽ 0. There is a vertical plane of mirror symmetry through the centre of the indenter. A friction coefficient of λ ⫽ 0.75 was measured from the critical angle at which a PU foam block slid from an inclined smooth metal plate. However, to obtain stable solutions, the indenter lower surface was tied to the foam upper surface (later, dynamic FEA allowed sliding at this surface, with almost the same predictions). For contact between the other metal surfaces and the foam, λ ⫽ 0.75 was used. The predicted shapes of the deformed elements (Fig. 6.5a) compares well with photographs of a grid on the foam front surface (Fig. 6.5b). At the support table surface, the grid lines in the deformed foam are


20

Stress (kPa)

2 15

1

10

5

0

Figure 6.6

0

0.1

0.2

0.3

0.4 Strain

0.5

0.6

0.7

0.8

Predicted plane-strain indentation stress vs. average compressive strain, (dashed curves) for the Ogden models of Figure 6.2, compared with loading–unloading data for PU38 foam (solid curves) (Mills and Gilchrist, 2000b).

not vertical, because of the friction-induced shear stresses at the foam–metal interface. High shear strains at the sides of the indenter allow the foam to wrap itself around the vertical indenter sides, and the sides of the foam rise from the support table. The initially vertical grid lines bulge slightly outwards at the sides of the indenter, as the foam expands laterally from under the indenter. – (indentation The variation of the average indentation stress σ force/indenter area), for PU38 foam, with compressive strain is shown in Figure 6.6. The prediction is close to the experimental unloading data, but about 5 kPa below the experimental loading data, for strains ⬍0.5. At strains ⬎0.6 the prediction rises towards the experimental loading data. The indentation stress is underestimated because the model parameters underestimate the initial plateau part of the foam compressive loading curve by about 5 kPa (Fig. 6.2). The experimental hysteresis, due to the polymer (non-linear) viscoelasticity in the bent foam edges, cannot be predicted by a hyperelastic model. The indentation stress ratio H is defined as H ≡

σ(ε) σU (ε)

(6.14)

where σU is the stress for uniaxial compression to the same nominal strain ε. The model predicts a peak H at a strain 艑 0.4, an underestimate by about 20% (Table 6.1). At low strains, the main stressed


Table 6.1

127

Peak indentation force ratio H in plane-strain indentation tests Friction coefficient

Experiment PU38

0 0.75 Tied surface

1.85, 1.95

FEA (N ⫽ 2) 1.58 1.59 1.60

region is directly below the indenter, and H 艑 1.4. At medium strains, the foam block is highly distorted, and strains in the foam to the side of the indentation cause H to increase. At high strains, the highly stressed foam directly below the indenter transmits the majority of the load, so H falls towards unity. The assumed friction coefficient λ has little effect on the predicted H value (Table 6.1), but when λ ⫽ 0 the sides of the deformed foam block rise more than when λ ⫽ 0.75.

6.3

Crushable foams The term crushable implies permanent plastic deformation or fracture of a compressed foam. Polystyrene (PS), polypropylene (PP), polyethylene (PE), and some rigid PU foams fall into this category. The yield stress of these closed-cell foams increases with deformation. This hardening mechanism is due to the compression of the cell gas, so relates to the imposed volumetric strain. In contrast, metal hardening is a function of the cumulative strain history – bending a copper wire back and forward several times hardens it by the buildup of dislocation structures, although the strain of the straightened wire is zero. FEA material models for crushable foams have the limitations of assuming material isotropy, and that the onset of yielding occurs in a similar way under varying stress states. They also ignore fracture; element elimination can be used to simulate fracture (Chapter 18).

6.3.1 Yield surfaces It is usual to assume that plastically deforming metals obey the von Mises yield criterion. The von Mises equivalent stress σe is defined in terms of the principal stresses σi by

σe2 ≡

1⎡ (σ ⫺ σ2 )2 ⫹ (σ1 ⫺ σ3 )2 ⫹ (σ2 ⫺ σ3 )2 ⎤⎥ ⎦ 2 ⎣⎢ 1

(6.15)

128 Polymer Foams Handbook If the value of σe exceeds a critical value, equal to the uniaxial compression yield stress, yielding will occur. The hydrostatic pressure component of the principal stresses, given by p⫽

1 (σ 3 1

⫹ σ2 ⫹ σ3 )

(6.16)

only causes a small elastic densification (the convention used is that compressive stresses are positive). Hence, if p is subtracted from the principal stresses, the deviatoric components of the principal stresses, σ1⫺p, σ2⫺p, and σ3⫺p, determine whether yielding occurs. This approach is justified since yielding occurs in metals by dislocation motion, which causes no volume change. The default stress contour map in ABAQUS is of the von Mises stress σe. A yield surface is a surface, in ‘stress space’ with axes σ1, σ2, and σ3, that describes the combinations of principal stresses that cause yielding. For the von Mises yield criterion, sections of this surface in planes containing the hydrostatic axis (the line σ1 ⫽ σ2 ⫽ σ3) consist of two parallel lines, equispaced from the hydrostatic axis. FEA requires a geometric rule that relates the plastic shape change to the co-ordinates of the stress state that caused yielding. For the von Mises yield criterion, the plastic strain increment is in a direction normal to the yield surface, at the yield point.

6.3.2 Crushable foam model in ABAQUS The crushable foam model uses a yield surface that is a modification of that for porous materials such as clays. The von Mises equivalent stress σe depends on the hydrostatic pressure component p according to ⎞2 ⎛ ⎞2 ⎛ ⎜⎜ p ⫺ 1 (p ⫺ p )⎟⎟ ⫹ ⎜⎜ aσe ⎟⎟ ⫽ a2 t ⎟ ⎜⎜⎝ b ⎟⎟⎠ ⎜⎝ ⎟⎠ 2 c

(6.17)

The yield surface in the p vs. σe plane is an ellipse with half-axes a and b in the p and σe directions, respectively. It intercepts the p-axis at ⫺pt and pC0, respectively, the initial yield stresses in hydrostatic tension and compression. Figure 6.7 is drawn for the special case where the centre of the ellipse is at zero hydrostatic stress. The model’s three main sections (elastic, initial yield, and hardening) are dealt with separately. 6.3.2.1 Elastic ABAQUS, from version 6.2 onwards, abandoned a porous elastic model that was developed for soils, for conventional elasticity. Masso-Moreu and Mills (2002) used the parameters Young’s modulus


129

0.5 C σc

Uniaxial compression

0.4 Deviatoric stress (MPa)

B σC0

0.3

0.2

0.1

0 −0.2

Figure 6.7

−pt −0.1

pC0 0 0.1 Hydrostatic stress (MPa)

0.2

0.3

Sections of the initial and hardened yield surface for XPS 35 foam, in the plane containing the hydrostatic axis. The surfaces have mirror symmetry about the horizontal axis (Gilchrist and Mills, 2001).

E ⫽ 10 MPa and Poisson’s ratio ν ⫽ 0.1 to match the tensile elastic response of XPS 35 foam. 6.3.2.2 Initial yield The shape of the initial yield surface (Fig. 6.7) is determined by the values of three initial yield stresses (or, as the compressive yield stress is defined in the hardening response, by the ratios of the second and third stresses to the first): 1. Uniaxial compression σC0 2. Hydrostatic compression pC0 3. Hydrostatic tension pt. The first stress is easily measured, there is sometimes data for the second, but it is necessary to guess the third (a sudden reduction in pressure from atmospheric to a vacuum would apply a hydrostatic tension of 100 kPa, which is insufficient to cause most rigid foams to yield). Section 6.3.4 explores alternative tests for evaluating the yield surface. Gilchrist and Mills (2001) used parameters in ABAQUS 5.8 equivalent to pC0 ⫽ 0.14 MPa, σC0 ⫽ 0.29 MPa, and pt ⫽ 0.15 MPa,

130 Polymer Foams Handbook for XPS 35 foam. The estimated pC0 was confirmed by measurements later (Masso-Moreu and Mills, 2004). 6.3.2.3. Hardening Hardening is assumed to be a function of the compressive inelastic volumetric strain εVi defined by ⎛ V ⎞⎟ εV ⫽ ⫺ln ⎜⎜⎜ ⎟⎟ ⎜⎝ V0 ⎟⎠

(6.18)

Tabular data of the uniaxial compression stress vs. the true compressive strain can be entered. The data is used to compute the horizontal diameter of the yield surface pc ⫹ pt as a function of the volume strain. The form of the hardening function should preferably be linked to the foam deformation mechanisms. The model for uniaxial compression of closed-cell foams (Chapter 11) relates the stress σC to the engineering strain ε by σC ⫽ σC0 ⫹

p0 ε 1⫺ ε ⫺ R

(6.19)

where σC0 is the initial yield stress, R is the foam relative density, and p0 is the effective absolute gas pressure in the undeformed cells. Gilchrist and Mills (2001) used σC0 ⫽ 0.29 MPa and p0 ⫽ 0.15 MPa to create a table of uniaxial compression data for XPS 35 foam. If Poisson’s ratio is zero, εV ⫽ ⫺ln λ, where λ is the extension ratio in the compression direction. When its volume reduces, the foam hardens and the co-ordinate pt at the left of the ellipse remains fixed, while that at the right moves to pC. The ellipse is magnified, while the ratio of a to b remains constant. Figure 6.7 shows the loading path for uniaxial compression, which intercepts the initial yield surface where σe ⫽ σC0, and the hardened yield surface at σC. However, in a simple shear test in which the volume is constant, the foam does not harden, so the test cannot proceed from point B to C along the deviatoric stress axis.

6.3.3 Response of crushable foam model in simple deformations The crushable foam model should be validated by comparison with the mechanical responses of low-density polymeric foams. Since the ABAQUS source code was not available, Gilchrist and Mills (2001) evaluated the model responses in loading/unloading ‘thought experiments’ on single or multi-element cubes. The following phenomena were revealed.


131

0.8

Compressive stress (MPa)

0.6 Data 0.4 FEA

0.2 0 −0.2 −0.4

Figure 6.8

0

0.1

0.2

0.3

0.4 Strain

0.5

0.6

0.7

0.8

Uniaxial compression loading–unloading response: FEA prediction of crushable foam model compared with impact data for EPS of density 60 kg m⫺3 (Gilchrist and Mills, 2001).

In uniaxial compression (Fig. 6.8) metal-like reverse yielding is predicted, requiring a high tensile stress to force the foam to return to its original dimensions. This is quite different from the experimental unloading response. The inelastic Poisson’s ratio is predicted to be ⬍0.03 when the compressive strain is ⬎0.2. The foam is predicted to yield in tension at a slightly smaller stress than in compression, and further yield occurs without hardening. Experimentally, the tensile yield stress exceeds the compressive one, and there is very little ductility before fracture. The predicted simple shear response has no hardening after yield (Fig. 6.9), because there is no volume change. However, using the impact equipment described in Section 6.3.4.3, an EPS foam of density 60 kg m⫺3 was found to harden when sheared. On unloading, when the shear direction is reversed, the shear hardening disappears. In closed-cell foams, cell faces align towards the principal tensile strain direction, so there is hardening in a tensile test. Since failure occurs at a relatively low tensile strain, the tensile response can be approximated as linearly elastic, rather than elastic–plastic.

6.3.4 Experimental data 6.3.4.1

Hydrostatic compression experiments

The foam model in ABAQUS 5.8 required hardening data obtained in hydrostatic compression, so this test has been performed, although


Shear stress (MPa)

0.6

0.4

Data

0.2

FEA

0

⫺0.2 0

Figure 6.9

0.1

0.2

0.3

0.4 0.5 Shear strain

0.6

0.7

0.8

Simple shear loading–unloading response: FEA prediction of crushable foam model compared with impact data for EPS of density 60 kg m⫺3 (Gilchrist and Mills, 2001).

few foam products are subjected to hydrostatic compression. Later versions of ABAQUS use uniaxial compression hardening data, which are converted into hydrostatic pressure vs. volume strain, within the programme. For low-density closed-cell foams, made from low modulus polymers, post-yield hardening is mostly caused by gas compression alone, so is a function of the volumetric strain. However, for foams made from higher modulus polymers, there is also a polymeric contribution that depends on the principal tensile strain. Such a contribution is not part of the ABAQUS foam model. Hydrostatic compression testing requires relatively complex specimen preparation (covering the foam with an impermeable but flexible membrane preventing liquid ingress) and equipment (instrumentation inside a fluid-filled pressure chamber). A number of phenomena make the data less than ideal for use in modelling foam impacts. One is the effect of air phase in the foam. Maji et al. (1995) tested rigid PU foams at a slow strain rate. Air lost from the foam, by diffusion through cell faces or from cell face fracture, can pass through the silicone–rubber water-repellent membrane and dissolve under pressure in water. In Nusholtz et al. (1996) experiments, air was free to pass to the atmosphere, so the mass of air in the foam was not constant. Zhang et al. (1997) tested expanded polypropylene (EPP) foam, allowing air to escape through a vent, but did not consider how the removal of air from the inter-bead channels affected the result. Bilkhu et al. (1993), who hydrostatically tested EPS foam of 40 kg m⫺3 density, found that initial yield occurred at about 13% volume strain, much higher than the 4% strain in uniaxial tests.


133

500

d

Relative pressure (kPa)

400

300 c b 200 a

100

0

0

0.5

1

1.5

2

εv/(1−R−εv)

Figure 6.10

Pressure vs. volume strain measure for EPP43, for application of different maximum pressures, followed by pressure release (Masso-Moreu and Mills, 2004).

The strain rate in these tests is usually low, with incremental increases in pressure. Hence the data may need correction for use in impact simulations. Currently, the only data gathered at reasonably high speeds is that of Masso-Moreu and Mills (2002), who used air pressure changes in the order of a few seconds, during which there was insignificant air diffusion from the foam. Figure 6.10 shows the response of EPP foam of density 43 kg m⫺3 plotted to suit the equation pc ⫽ pC0 ⫹

pcell e V 1 ⫺ eV ⫺ R

(6.20)

where pc is the yield stress in hydrostatic compression, pC0 the initial value, pcell the effective absolute gas pressure in the foam cells, and eV the engineering volume strain defined from the initial foam volume V0 and deformed volume V as eV ⫽

(V0 ⫺ V ) V0

(6.21)

The parameters that fit the loading response are pC0 ⫽ 169 kPa and pcell ⫽ 151 kPa, while for unloading, pC0 ⫽ ⫺7 kPa and pcell ⫽ 146 kPa.

134 Polymer Foams Handbook If data determined at low strain rates is used for rate-dependent PP or PU foams, the predicted hardening for uniaxial compression impact could be in error due to the strain rate factor and a different hardening rate in uniaxial compression. In order to compare hardening in uniaxial and hydrostatic compression, an appropriate strain measure must be chosen. Usually uniaxial data was obtained without determination of the lateral strains, so the latter were assumed to be zero, and the volume strain calculated. Nusholtz et al. (1996) found for EPP foam of density 80 kg m⫺3, the uniaxial compressive stress was 60% higher than the hydrostatic pressure needed to achieve the same true volume strain. This may be due to anisotropy. Zhang et al. (1997) fitted three data points (simple shear, uniaxial compression, and hydrostatic compression) for EPP of density 49 kg m⫺3 with equation (6.17), to obtain a yield surface with ellipticity b/a ⫽ 3.8. The yield surfaces suggest that the hydrostatic tensile yield stress increases as the volume strain increases. Deshpande and Fleck (2001) carried out triaxial tension experiments using three orthogonal actuators and a complex shaped specimen in which only the central portion was under triaxial tension. They did not examine the strain distribution in the deformed specimen, so did not confirm that there was yielding under triaxial tension. If there was plane-strain uniaxial tension yielding in the outer regions of the specimen, the measured value (Fig. 6.13) would be a lower bound for the triaxial tension yield stress. 6.3.4.2

Multi-axial stress experiments

Maji et al. (1995) placed rigid PU foam specimens (of a range of densities) in a water-filled chamber. The water was separated from the rigid PU foam by a flexible rubber membrane, which was clamped to the upper and lower platens of a compression testing machine. A constant water pressure p was applied, and then the crosshead moved to apply a uniaxial compressive stress σc in the vertical direction. The principal compressive stresses in the horizontal directions were σ2 ⫽ σ3 ⫽ p

(6.22)

They assumed that the principal stress in the vertical direction was σ1 ⫽ p ⫹ σc

(6.23)

Presumably the load cell was zeroed after the application of the hydrostatic pressure and before crosshead motion. The conditions along the metal–foam interface are unknown; friction at this interface would cause a non-uniform distribution of σ1. Measurement of the pressure distribution on the piston face would show whether equation (6.22) is valid. The foams were anisotropic, with a higher uniaxial compressive strength along the rise (R) direction (which


135

they called the parallel direction) than along the perpendicular (P) directions, but transversely isotropic. The stress σ1R for compressive yielding along the R direction was 0.75 ⫾ 0.03 MPa for the 80 kg m⫺3 density foam, independent of the confining pressure, whereas σ1P was constant at 1.1 ⫾ 0.1 MPa. In a hydrostatic compression test, yielding occurred at a pressure of 0.76 ⫾ 0.03 MPa. Deshpande and Fleck (2001) carried out similar experiments on PVC foams, using hydraulic fluid instead of water. 6.3.4.3

Shear plus compression experiments

Mills and Gilchrist (1999) performed oblique impact tests on EPP foams using the equipment shown in Figure 6.11. The lower surfaces of rectangular foam blocks (50 mm long, 20 mm wide, and 20 mm tall) were fixed to flat aluminium plates with epoxy resin. Their upper surfaces, inclined at angle θ ⫽ 15°, 30°, or 45° to the horizontal, were impacted from heights in the range 0.25–1 m with a vertically falling V-shaped striker. The impact surface was covered with 120 grade SiC paper to avoid slip. A triaxial quartz load cell was mounted below one of the aluminium plates. The striker acceleration was integrated twice, and used with the free-fall velocity to obtain the vertical deflection X of the top of the foam block (see Chapter 5). The compressive strain ε and shear strain γ in the foam are related to X by ε⫽

X cos θ h

γ ⫽

X sin θ h

(6.24)

Accelerometer Aluminium block constrained to move vertically

Foam

SiC paper Adhesive

Foam

h

FN

FT

Triaxial force cell Fixed aluminium support frame

Figure 6.11

A vertically falling mass impacts two foam blocks supported on a V-shaped rigid surface, with a triaxial quartz load cell below one foam block (redrawn from Mills and Gilchrist, 1999).

136 Polymer Foams Handbook 1 C

0.6

Relative stress (MPa)

Stress (MPa)

0.8

S

0.4 0.2 0 −0.2

3.5

45°

Y 0

(a)

Figure 6.12

20

40 60 Strain (%)

X 80

3 45° Oblique

2.5 2

(b)

Simple shear

1 0.5

100

Tensile

1.5

1

1.05 1.1 1.15 1.2 1.25 1.3 1.35 Elongation ratio

(a) Shear S and compressive C stresses vs. shear or compressive strains, for oblique impact tests, at initial strain rate 121s⫺1, on EPP of density 43 kgm⫺3 at 45° inclination. (b) Shear stresses, relative to the initial collapse value, plotted against the principal tensile extension ratio (Mills and Gilchrist, 1999).

The assumption of no slip at the foam upper surface was confirmed by the unloading response. The deformation is plane-strain. The most tensile principal elongation ratio in the foam was computed by the equivalent of equation (5.18), as a function of the shear strain; it becomes quite large in pure shear tests, but is insignificant for the 15° test. Figure 6.12a shows both compressive and shear stresses harden in a test when θ ⫽ 45°. As the foam is strain-rate dependent, to allow comparison with static tests, the shear stress was divided by the initial shear yield stress. This relative shear stress increases with the principal extension ratio of the foam (Fig. 6.12b) for a number of test types. However, the 45° oblique impact data diverges for elongation ratios ⬎1.2. The shear hardening effect must be polymeric in origin, as there is no volume change in a simple shear test. In the deformed EPP, some of the biaxially oriented cell faces are stretched. Since biaxially oriented PP film shows significant hardening after the initial yield in a tensile test, this deformation mode in some EPP cell faces causes the hardening of the sheared PP foam. For EPP foam of density 43 kg m⫺3, the compressive stress was fitted with equation (6.19); the constants are given in Table 6.2. The initial yield stress σ0 at impact strain rates is smaller for the 45° than for the 15° oblique test, but the gas pressure hardening parameter p0 remains approximately constant. Since p0 exceeds 100 kPa, some of the compressive hardening must be due to a polymeric mechanism. Since the impact values of σ0 are significantly larger than the values from slow tests, the EPP foam is rate dependent. Unloading mainly changes the parameter σ0 in contrast with the prediction of the FOAM model (Fig. 6.8).


Table 6.2

137

Compressive response of PP bead foam of density 43 kg m⫺3 Test

Strain rate (s⫺1)

Direction

σ0 (kPa)

Uniaxial Uniaxial Uniaxial 15° oblique 45° oblique

2.2 ⫻ 10⫺2 2.2 ⫻ 10⫺2 78 120 120

Loading Unloading Loading Loading Loading

200 ⫺20 352 390 215

p0 (kPa) 159 159 135 151 148

Deviatoric stress σ = S −T (MPa)

5

H100

0

H200

−5 −5

0

5

S + 2T Mean stress σm = (MPa) 3

Figure 6.13

Yield surfaces of PVC foams of densities 100 and 200 kg m⫺3.The ellipses represent equation (4.20) and the dashed lines represent yield by buckling (Deshpande and Fleck, 2001 © Elsevier).

6.3.4.4

Yield surface data

Deshpande and Fleck (2001) determined the yield response of two densities of Diab ‘PVC’ foam (see Chapter 18) using a range of loading types. They used a yield criterion 2 ) ⫽ σ2 0.9(σe2 ⫹ σm y

(6.25)

where the mean stress σm is ⫺p of equation (6.17) and σy is the yield stress in uniaxial tension or uniaxial compression. This ellipse is nearly circular (Fig. 6.13); its ellipticity of b/a ⫽ 1.1 is significantly smaller than those used for XPS (Masso-Moreu and Mills, 2002) or


Cauchy tensile or compressive stress (MPa)

3

2.5 Uniaxial tension 2 Uniaxial compression 1.5

1

Parallel to rise direction 0

Figure 6.14

Perpendicular to rise direction

0.5

0

0.02

0.04 0.06 0.08 0.1 True tensile or compressive strain

0.12

0.14

Uniaxial compressive and tensile responses of PVC foam of density 100 kg m⫺3 in both the rise and perpendicular directions. A construction for the initial yield point • in uniaxial tension is shown (Deshpande and Fleck, 2001 © Elsevier).

EPP (Zhang et al., 1997). Note that the sign convention for positive hydrostatic stress is opposite to that in Figure 6.7. The ellipse was combined with a ‘cut-off’ criterion, shown by the dashed lines, that yielding would occur by buckling if the maximum compressive principal stress exceeded a critical value (2.4 MPa for the 100 kg m⫺3 density and 6.4 MPa for the 200 kg m⫺3 density foam). However, the PVC foam initial yield stress is ⬃20% higher in the rise direction of the foam sheet than in a perpendicular direction (Fig. 6.14). This anisotropy is not considered in the criterion. Also, the definition of the initial yield stress varies with test type; when there is a yield drop, the initial maximum stress is used, but in uniaxial tension tests, the intersection of the initial elastic and linear hardening lines is used. The uniaxial tension data ends when foam fracture occurs. Hence, the ‘yield’ surface, described by equation (6.25), includes some data for semi-brittle failure. As the yield surface has not been implemented in ABAQUS, it is not possible to check its performance against the crushable foam criterion.

6.3.5 Validation of FEA models The best way to validate an FEA material model is to compare the predicted responses for products with complex stress fields with

Chapter 6 Finite element modelling of foam deformation Mirror symmetry

0.25 0.2 0.1

139

2.5 Experiment

FEA

0

Force (kN)

2 1.5 1 0.5 0 (a)

Figure 6.15

(b)

0

5

10 15 20 25 30 Foam deformation (mm)

35

40

(a) Shape of PS35 foam block, predicted by dynamic FEA, 2.8 ms after impact of a 39 mm diameter cylinder, with contours of maximum principal tensile stress (MPa). (b) Predicted force vs. distance (unpublished) compared with data for PS35 foam (Gilchrist and Mills, 2001).

experimental data. The material model parameters should be measured, rather than empirically adjusted, to improve the predictions. If only a single product shape is simulated, parameter adjustment can lead to reasonable agreement with test data. Two examples illustrate some pitfalls in validation. Puso and Govinjee (1995) considered the impact on a block of PU foam, using the indentation force deflection (IFD) geometry. Since the 150 mm indenter diameter was comparable with the 254 mm block width, and the low foam thickness (62 mm), FEA of such geometry suggests a H value equation (6.4) not ⬎⬎ 1, that is the foam under the indenter is approximately under uniaxial compression. They characterised the PU foam in uniaxial compression and predicted the average stress for indentation to within 10%. However, the stress field of this validation experiment was not demanding. Gilchrist and Mills (2001) carried out plane-strain deformation impact experiments on an XPS of nominal density 35 kg m⫺3. A block of EPS of height 75 mm, width 118 mm, and depth 70 mm was impacted with the side of a cylinder of diameter 39 mm, at 5 m s⫺1. A square grid, with 10 mm spacing, was marked on the front face of the specimen. Their static FEA was repeated recently using dynamic FEA. A vertical mirror symmetry plane through the centre of the foam reduced the problem size. The predicted foam geometry 2.8 ms after impact is shown in Figure 6.15a. At this stage, the tensile principal stress on the foam upper surface, just outside the contact region, reached a maximum of 0.25 MPa, due to localised bending. High speed video (Fig. 6.16) shows surface cracks initiated at the same site


5 ms

3 ms

(a)

(b)

11 ms

7 ms (c)

(d)

Figure 6.16

Video frames of XPS 35 foam indented with a 39 mm diameter cylinder, at the times indicated (Gilchrist and Mills, 2001).

by 3 ms, and grew to lengths of 10–20 mm. The experimental indentation force is a non-linear function of the indentation depth, with negative curvature (Fig. 6.15b). It is about 0.3 kN higher than the prediction for most of the curve. As the cracking would reduce the force, this suggests that the FEA is underestimating some aspect of stress field, such as the shear stresses at the sides of the indentation. The predicted energy recovery on unloading was smaller than that observed, a fault of the material model discussed in Section 6.3.2.

6.4

Dynamic FEA (explicit) 6.4.1 Computing issues Dynamic FEA, although better suited to impact problems, introduces a number of computing issues. The explicit method estimates the


141

stress state at a series of time intervals. The time interval ∆t between the computations must be smaller than the time for a sound wave to cross the smallest element. The foam density and Young’s modulus control the speed of sound in the foam. In the simulation of impact tests on large foam blocks, the overall simulation time is of the order of 20 ms and the ∆t values for the relative large elements are reasonable. However, foam products with complex geometry (such as bicycle helmets) require small elements in regions of high curvature, so the time interval can be extremely small. Meshes may need to be amended to replace small elements in order to obtain solutions in a reasonable time. Further issues arise when foam contacts a rigid (infinitely stiff) surface; for instance a headform can be treated as rigid (Chapter 14) to reduce the problem size. It is not possible to use kinematic contact conditions with ‘hard’ contact (the bodies cannot overlap in space). The alternative ‘penalty’ contact conditions introduce a fictitious elastic contact spring at the interface, specifying how the over-closure of the contact surfaces changes as a function of pressure. Consequently, there will be some overlap of the foam and the rigid surface. This can be overcome by using hard contact with an elastic body, such as a steel plate, at the expense of increasing the problem size. Simulation will cease if any element inverts (develops a negative volume); to overcome this problem, a distortion control parameter sets a lower limit (typically 0.1) on the relative height of the element. Another ‘trick’ is to give the elements a bulk viscosity, with linear and quadratic terms. The former damps high frequency ringing, while the latter smears shock waves across several elements and prevents elements from collapsing. Default values of these parameters are often used. A damping factor can also be introduced to simulate material damping. A better alternative is to consider the material linear viscoelastic response using a generalised Maxwell model (see Chapter 19).

6.4.2 Simulation of foam compressive impact tests A typical uniaxial compressive impact test on EPS was simulated. A 100 mm cube of EPS of density 35 kg m⫺3 was struck by a flat-faced striker, of mass 5 kg, at 5 m s⫺l. Two FEA outputs, the striker acceleration and the reaction force on the rigid support table, were processed as described in Chapter 5, to generate the foam stresses at the striker stress and table interfaces. Figure 6.17a shows that the stresses oscillate, especially in the initial elastic region, in a plot against the foam strain. In the post-yield part of the curve, the two stresses oscillate out of phase, due to the transmission and reflection of sound waves through the foam. As the material model had a smoothly increasing stress–strain curve, such oscillations should be

142 Polymer Foams Handbook 0.6

Stress (MPa)

0.5 0.4 0.3 0.2 0.1 0

0

0.1

(a)

Figure 6.17

0.2 0.3 0.4 0.5 Compressive strain

0.6

0.7 (b)

Dynamic FEA of a 5 m s⫺1 impact of an XPS 35 100 mm cube. (a) Stress–strain curves: solid line striker stress, dotted line table stress. (b) Plastic strain contours (range 0.43–0.51) at time 10 ms when the mean total strain was 0.43 (Mills, unpublished).

ignored when impact compression experiments are analysed to extract the foam stress–strain response. Figure 6.17b shows that the vertical component of plastic strain is slightly inhomogeneous, but the foam deforms almost uniformly. When impact tests were simulated for open-cell flexible PU foams, (Mills, 2006) a different phenomenon arises due to inertia of the relatively dense foam, its low elastic modulus, and the low sound wave velocity. The model consisted of one-quarter of a foam cube of side 100 mm, with two vertical symmetry planes running through the centre point, supported on a rigid flat surface, and impacted at 3.5 m s⫺1 by a rigid flat-faced striker of total mass 2 kg. Figure 6.18a shows the striker and table stresses vs. strain, predicted by dynamic FEA. The initial striker-to-foam-top impact creates a pressure pulse which travels down through the foam. Figure 6.18b shows that, after 2.4 ms, the upper part of the specimen is compressed and under a nearly uniform stress of 6 kPa, while the lower part is almost stress free. With increasing time, the higher stress region propagates to the lower surface of the specimen, then a new, yet higher stress region forms near this surface and propagates upwards through the foam. The pulses appear as steps on the loading part of the stress–strain curve. They have effectively disappeared by the time the foam is at maximum strain. A initial step of 5 kPa height, with some small oscillations, was observed in the striker stress vs. foam strain graphs derived from impacts on 100 mm thick blocks of PU chip foam (Lyn and Mills, 2001). Dynamic FEA often allows higher strain simulations than static FEA. For example, using static FEA, Masso-Moreu and Mills (2003)


143

50

Striker

5 3

4 2

1

Stress (KPa)

40

30

20 Table

Symmetry plane

10

0 Rigid table

(a)

Figure 6.18

(b)

Striker

0

0.1

0.2


0.7

0.8

(a) Model of one-quarter of a foam cube after 2. 4 ms with contours of the vertical compressive stress component (kPa). (b) Predicted compressive stress–strain curves for a cube of foam: solid curves for standard FEA and dashed curves for the table stress and striker stress in dynamic FEA (Mills, 2006).

had to approximate the geometry of truncated XPS pyramids as axisymmetric solids to achieve large strains. However, with dynamic FEA, the correct geometry could be modelling successfully.

Summary Hyperelastic FEA models for flexible foams, with parameters that describe a range of test data types, can be used to analyse complex strain fields such as those in indentation tests. To model hysteresis or rate-dependent behaviour, viscoelastic behaviour can be associated with a hyperelastic model in ABAQUS Explicit; some examples are given in Chapter 19. The response of the crushable foam model in ABAQUS differs from that of low-density XPS, especially when the foam is unloaded, or during shear or tensile deformation. The foam acts as if an elastic volumetric hardening mechanism (due to cell gas pressure) was in parallel with a plasticity mechanism (that hardens in simple shear, but not in compression). The foams are softer in unloading than loading, because cell faces have formed plastic hinges during compressive loading. FEA predictions are approximations because: (a) Many rigid foams are slightly anisotropic, while the FEA is for isotropic materials. (b) The crushable foam model has the same hardening in uniaxial and hydrostatic compression. If the former is used as material

144 Polymer Foams Handbook data, the hardened yield surface expands excessively for stress states close to hydrostatic compression. (c) The ellipticity of the initial yield surface varies from 1.1 to 3.5 for different polymers. Given the similar microstructures, the values should be more consistent. However, the amount of tensile ductility varies with the polymer and the foam anisotropy varies with the processing. It is hoped that new material models will combine the behaviour of hyperelastic foams and crushable rigid foams.

References ABAQUS (2004) Users Manual Version 6.4, Hibbitt, Karlsson and Sorensen, Warrington, UK. ABAQUS (1998) Standard Examples Problem Manual, Hibbitt, Karlsson and Sorensen, Warrington, UK. Bilkhu S.S., Founas M. & Nusholtz G.S. (1993) Material modelling of structural foams in FEA using compressive uniaxial and triaxial data, SAE Trans. 102, 547–565. Davies O.L., Gilchrist A. & Mills N.J. (2000) Seating pressure distributions using slow-recovery PU foams, Cell. Polym. 19, 1–24. Deshpande V.S. & Fleck N. (2001) Multiaxial yield behaviour of polymer foams, Acta Mater. 49, 1859–1866. Gilchrist A. & Mills N.J. (2001) Impact deformation of rigid polymeric foams, experiments and FEA modelling, Int. J. Impact Eng. 25, 767–786. Lemmon T., Shiang T.Y. et al. (1997) The effect of insoles in therapeutic footwear – a finite element approach, J. Biomech. 30, 615–620. Lyn G. & Mills N.J. (2001) Design of foam crash mats for head impact protection, Sport. Eng. 4, 153–163. Maji A.K., Schreyer H.L. et al. (1995) Mechanical properties of PU foam impact limiters, J. Eng. Mech. 121, 528–540. Masso-Moreu Y. & Mills N.J. (2003) Impact compression of polystyrene foam pyramids, Int. J. Impact Eng. 28, 653–676. Masso-Moreu Y. & Mills N.J. (2004) Rapid hydrostatic testing of rigid polymer foams, Polym. Test. 23, 313–322. Mills N.J. (2006) Finite element models for the viscoelasticity of open-cell polyurethane foam, Cell. Polym. 25, 277–300. Mills N.J. & Gilchrist A. (1999) Shear and compressive impact of PP bead foam, Cell. Polym. 18, 157–174. Mills N.J. & Gilchrist A. (2000a) High strain extension of open cell foams, J. Eng. Mater. Tech. ASME 122, 57–63. Mills N.J. & Gilchrist A. (2000b) Modelling the indentation of lowdensity polymer foams, Cell. Polym. 19, 389–412.


145

Nusholtz G.S., Bilkhu S. et al. (1996) Impact response of foam, the effect of the state of stress, Stapp Conference, 95, SAE. Ogden R.W. (1972) Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids, Proc. Roy. Soc. London. A328, 567–583. Ogden R.W. (1997) Non-linear Elastic Deformations, Dover, New York. Puso M.A. & Govinjee S. (1995) A phenomenological constitutive equation for rigid polymeric foam, in Mechanics of Plastics and Plastic Composites, ASME, MD-68. Rivlin R.S. (1948) Large elastic deformations of isotropic materials, Part 1, Phil. Trans. Roy. Soc. Lond. A240, 459–490. Rusch K.C. (1970) Load compression behaviour of brittle foams, J. Appl. Poly. Sci. 14, 1263–1276. Schrodt M., Benderoth G. et al. (2005) Hyperelastic description of polymer soft foams at finite deformations, Tech. Mech. 25, 163–173. Setyabudhy R.H., Ali A. et al. (1997) Measuring and modelling of human soft tissue and seat interaction, SAE STP-1242, Progress with Human Factors in Automotive Design, 135–142. Warrendale, PA (USA). Shames I.H. & Dym C.L. (1985) Energy and Finite Element Methods in Structural Mechanics, Taylor and Francis. Zhang J., Lin Z. et al. (1997) Constitutive modelling and material characterisation of polymeric foams, J. Eng. Mat. Tech. Trans. ASME 119, 284–291.

Chapter 7

Micromechanics of open-cell foams

Chapter contents 7.1 7.2 7.3 7.4 7.5

Introduction Edge geometry and stiffness Regular polyhedral-cell models Elastic moduli of the Kelvin foam Compression of the Kelvin foam with uniform edges 7.6 FEA model of wet Kelvin foam 7.7 Irregular foam models 7.8 Anisotropic cell shapes 7.9 Non-linear polymer response 7.10 Strain localisation 7.11 Modelling edge touching 7.12 Comparison with experiment References

148 151 153 156 160 163 167 169 170 171 171 172 173


7.1

Introduction 7.1.1 Concepts and approaches Materials science relates the mechanical properties of foams to their geometry and the polymer mechanical properties. Foam micromechanics may also assist in the development of material models used in finite element analysis (FEA). Over the last 35 years many models have been proposed for the elastic moduli of open-cell polymer foams; some were reviewed by Hilyard (1982) and Kraynik and Warren (1994). However, there have been blind turnings, and some concepts have been superseded. Modelling has been influenced by Gibson and Ashby’s book (1988), and their concepts will be critically reviewed. We start by reviewing deformation mechanisms. The concepts which explain open-cell foam mechanics (Fig. 7.1) are then explained in order of increasing complexity. Air flow through open-cell foams usually does not contribute to the mechanical response, so its consideration is left to Chapter 8. The geometry of a model should be close to that of the open-cell foam; we consider here low-density flexible foams, but not open-cell

Foam feature

Theory

Edge

/ and J, second moments of edge area

Polyhedral cells (with uniform edges)

Elastic moduli of kelvin foam with beam edges Kelvin foam [001] compression using elliptic integrals

Full edge and vertex geomery

Wet Kelvin foam FEA of [001] and [111] compression Long-range buckling in Kelvin foam [001] compression

Figure 7.1

Irregular cell shapes

Voronoi models

Anisotropic cell shapes and variable cell sizes

CT scan converted to pixellated models

Concepts used to explain the mechanics of open-cell foams.

Chapter 7 Micromechanics of open-cell foams

149

polyethylene foams, in which only a few cell faces are holed, or polyurethane (PU) foams with relative density ⬎0.2 in which the cells are nearly spherical. The model should include all the observed edge deformation mechanisms and use appropriate polymer mechanical properties. Chapters 9 and 10 discuss some applications of these foams; they are mainly loaded in compression, but shear also occurs, for example at the sides of indentations.

7.1.2 Observations of cell deformation ‘Bulpren’ PU foam, of density 31 kg m⫺3, has cells elongated in the foam-rise direction (Zhu et al., 1997b). Individual cells, cut from the foam, were super-glued to two glass slides, (so the cells could not shear laterally) and then compressed. Figure 7.2 shows cells compressed in the rise direction and in a perpendicular direction. There is

(a)

(b)

(c)

(d)

Figure 7.2

10 mm

Individual PU foam cells: (a) undeformed, (b) compressed 28% along the rise direction, (c) undeformed, and (d) deformed 19% along a direction perpendicular to the rise direction (Zhu et al., 1997b).


0.4

Force (N)

Perpendicular 0.3

0.2 Rise 0.1

0

Figure 7.3

0

0.1

0.2

0.3

0.4

0.5

0.6

Force vs. compressive strain for compression of the two cells shown in Figure 7.2 (Zhu et al., 1997b).

a monotonic force increase for the latter, but a force maximum at about 20% strain for the former (Fig. 7.3). The force rises rapidly when the strain exceeds 50% and edges touch. In these experiments the top and bottom vertices, glued to a rigid plate, cannot rotate or move laterally; but the other vertices are not constrained. However, in a large piece of foam, every vertex is constrained by the deformation of the surrounding cells. Consequently, in the latter, the response and the deformed edge shapes differ from those in a single cell experiment. The longer cell edges, at an angle to the stress axis, tend to undergo high deformations first; they can deflect by 100% of their original height, since the bent edge can intrude into other parts of the cell. Shorter edges remain straight but reorient away from the stress axis (Fig. 7.2). There is some edge twisting. The deformation of edges, deep within a PU foam, was observed using confocal optical microscopy (Hamza et al., 1997). The stress– strain curves were similar to that in Figure 7.2. Some edges, parallel to the loading direction, became bent when the strain was about 15% and the stress had reached its initial plateau. Their bending increased when the strain reached 35%, whereas other edges were only slightly bent. At 75% strain, edges were seen to touch. Microscopy provides interesting details, but the significance of an individual event (the bending of a strut) on the overall foam stress is hard to gauge. Elliott et al. (2002) carried out CT tomography of a small region in the centre of a 25 mm cube of PU foam, with a wide range of cell sizes, as it was compressed. There was inhomogeneous deformation with collapse bands, and the dominant edge deformation mechanism was bending. Figure 7.4


151

C E

B A

D B A

C

E D

(b)

(a)

Figure 7.4

X-ray tomography images of a PU foam of density 19 kg m⫺3 at (a) 0% and (b) 23% compressive strains.The compression direction is vertical in the 6.8 mm by 6.8 mm by 1.7 mm thick sample (Elliott et al., 2002 © Kluwer).

compares the structure at 0% and 23% compressive strains. The large cells labelled A to E across the top of the figure have collapsed in Figure 7.4b, whereas a similar set of large cells across the lower part of the figure have much less deformation.

7.2

Edge geometry and stiffness Low-density foams have slender edges with Plateau border cross-sections. Their bending and torsional stiffness depend on their width, and width variation along the edges should be considered. However, until recently, models assumed that edges have constant width to simplify the analysis. The bending stiffness of an edge (or beam) depends on its second moment of area, I, defined by I =

∫

wy2 dy

(7.1)

where y is the distance from the neutral surface and w the beam width. The neutral surface runs through the centroid of the crosssection. Table 7.1 gives I values for a number of cross-sectional shapes; the value is independent of the plane of bending, except for the square section. The torsional stiffness of a beam depends on the polar second moment of area J of the cross-sectional plane. The


Edge moments of area (Warren et al., 1997) and Kelvin foam Young’s modulus (Warren and Kraynik, 1997) I A2

J A2

E ER 2

πr2

0.0796

0.1592

0.593

2

0.0833

0.1406

0.619

0.0923

0.1155

0.710

0.1338

0.0808

0.979

Edge cross-section

Area A

Circle Square Equilateral triangle Plateau border

b

3b 2 b2

(

4 3⫺π 2

)

calculation of J requires the solution of Poisson’s equation for a potential function φ of the cross-section ∂ 2φ ∂ 2φ ⫹ ⫽ 2Gθ ∂x 2 ∂y 2

(7.2)

where G is the shear modulus and θ the angle of twist per unit length of beam. Finite element packages such as ABAQUS can be used to solve a two-dimensional (2D), steady state, heat flow problem in a body with a constant heat generation rate. This is controlled by another form of Poisson’s equation; a boundary at a fixed temperature is the analogue of the free surface at the beam boundary, while the total heat stored is the analogue of the torsional stiffness (Mills, 2005). Table 7.1 gives the J values in terms of r, the radius of a circular edge cross-section, or b, the side of triangular or square cross-sections, or the breadth (cusp to cusp) of Plateau border cross-sections. Warren and Kraynik (1997) defined the half-edge tensile compliance as the displacement of the midpoint with respect to the vertex, divided by the tensile force acting on the edge. The torsional compliance is the angle of twist of the midpoint divided by the torque on the edge (Table 7.2). In some regular foam models, an edge, initially aligned with the compressive stress direction, may buckle elastically. Buckling is a change in deformation mode at a critical load or strain. For example, a column axially loaded in compression remains straight for small loads; above a critical load it bends. However, only a tiny percentage of cell edges in PU foams are subjected to axial compression (even if aligned with the compression axis, the end loads may not be axial); so edge buckling is extremely rare. Highly deformed edges, which have progressively bent and twisted, are sometimes incorrectly described as buckled. Edges, which are subjected to bending and twisting loads, should not be described as struts (structures that are axially loaded in compression).


Table 7.2

7.3

153

Compliances of half-edges Deformation mechanism

Compliance

Units

Stretching

CS ≡

L EA

m N⫺1

Bending

CB ≡

L3 3EI

m N⫺1

Torsion

CT ≡

L 2GJ

m⫺1 N⫺1

Regular polyhedral-cell models When a regular lattice model is analysed for a particular type of loading, it is only necessary to consider a representative unit cell (RUC): the deformation pattern repeats periodically at its boundaries. This may differ from the ‘crystal’ unit cell of the undeformed structure, which is the smallest unit, when repeated by translation, that fills space by regular packing. The choice of the RUC is not trivial, as small RUCs may rule out a particular deformation pattern. Since FEA programmes cannot impose rotational symmetry conditions on boundary surfaces, larger RUCs must be used with mirror or periodic symmetry boundary conditions.

7.3.1 Gibson and Ashby model Gibson and Ashby’s (1988) analyses are for a compressive stress acting in the [001] direction of a cubic lattice (the Miller indices [hkl] for direction have vector components (h, k, l) in a cubic lattice). The RUC of their widely quoted model (Fig. 7.5a) is relatively large with mirror symmetry planes where the edges are cut. However, its geometry is inappropriate for open-cell foams since: (a) edges meet at 90°, an angle not seen in polymer foams. (b) edges have square rather than Plateau border cross-sections. (c) vertices link three edges, whereas vertices in polymer foams link four (or more) edges. (d) the structure is only continuously connected in two, rather than three dimensions. Hence it has infinite anisotropy with zero properties in the one direction in which the structure is not continuous. For the elastic response, they suggested that only the horizontal edges H, initially aligned perpendicular to compressive stress direction, bend.

154 Polymer Foams Handbook B B 0.35 H

H H

3

V

V

L V

V 2

H

1

H

B

0.3 Normalised stress

H

H

0.25 0.2 0.15 0.1 0.05

H

0 B

(a)

0

0.1


(b)

0.6

V

3 Mirror plane

H

9 6

3

6

3

Contact 12

6 3

6

3 3

2

B

Mirror plane 9

1 63

(c)

Figure 7.5

(d)

(e)

Gibson and Ashby model for compression of an open-cell foam made from an elastic material: (a) undeformed full model, (b) predicted normalised stress vs. strain relationship, and (c) deformed shape of 1/8th model at strains of 17.5%, (d) 30%, and (e) 55%, with von Mises stress contours in MPa (Mills, unpublished).

Using an expression for beam bending compliance (as in Table 7.2), and the relationship between the foam relative density R and the edge dimensions, they concluded that the foam Young’s modulus EF was given by EF ⫽ C1EPR2

(7.3)

where C1 is an unknown constant and EP the polymer modulus. The normalised Young’s modulus E* is defined by E* ≡

EF EP R2

(7.4)


155

One task of foam modelling is to predict the value of E*. By comparison with experimental data, rather than by making a mechanics analysis, Gibson and Ashby inferred that C1 (or E*) ⬵ 1.0. The author’s unpublished FEA assumed that the model repeats with a spacing L between the complete cubes in the 1-axis direction, while it is free to contract in the 3-axis directions. The edge width of 0.1 and length of 0.9 correspond to a relative density R ⫽ 0.026 if there are further edges in the 1-axis direction. A mesh spacing of 0.025 was used with quadratic order elements of cubic shape; with this relatively coarse mesh the results are accurate to about 5%. To obtain the deformation pattern shown in Gibson and Ashby’s figures, the RUC in Figure 7.5a must have 12, 23, and 31 mirror symmetry planes through a point at its centre. Hence the RUC used for FEA was 1/8th of the size of that in Figure 7.5a. The horizontal edges marked H bend easily and the vertical edges marked V bend at small foam strains, whereas Gibson and Ashby’s diagram shows them remaining straight. The foam normalised Young’s modulus E*001 ⫽ 0.78 is reasonable, but it is much lower for loading in a direction at an angle to the cube axes. In their model for the compressive collapse of elastomeric foams, Gibson and Ashby used Euler’s formula for the critical compressive force on an axially loaded column of length L PC ⫽ k

EI L2

(7.5)

where the constant k is related to the end constraint conditions. They applied this to the vertical edges V in Figure 7.5a and predicted that the elastic collapse stress was given by σ*el ⫽ C4EPR2

(7.6)

where C4 is an unknown constant. They refer to the elastic collapse stress as a plateau stress and label the region of PU stress–strain curves (where the stress is rising slowly) as a plateau. However, a plateau implies a constant stress for part of the compressive response, while equation (7.6) is only for the onset of buckling. Both large deformation Elastica methods (Section 7.5.2) and FEA predict that the post-buckling load on an axially loaded column increases with deflection. FEA predicts that the vertical edges B and V in the model buckle by bending in the 12 planes above 20% strain (Fig. 7.5d), at a normalised stress, defined by σ* ≡

σ EP R2

(7.7)

156 Polymer Foams Handbook of 0.12. At this point, the slope of the stress–strain graph suddenly decreases. The stress applied to the buckled structure increases up to 52% strain, where the first edge-to-edge contact causes a slope increase (hardening) in the stress–strain curve. Further edge-to-edge contacts cause a large slope above 60% strain. Gibson and Ashby do not attempt to use the model to predict the onset of edge-to-edge contact; they just say it happens at a high strain, causing the stress–strain graph to become near vertical. If one or more mirror symmetry planes are removed from the RUC, or an RUC with two or more cells in a vertical direction is analysed, the buckling mode differs, as does the stress–strain response. Hence the response in Figure 7.5b only relates to a small RUC. As the response depends on the RUC size, the model cannot represent foams containing many cells. It is a semi-quantitative mechanics analogue for foam behaviour, similar to spring and dashpot analogue models for viscoelastic materials (Chapter 19). Most readers do not trouble analysing Gibson and Ashby’s model, so are misled into accepting the claimed foam responses, and hence the assumed deformation mechanisms. Better models, with geometries closer to that of PU foams, will be explored in the following sections, and their quantitative predictions compared with experimental data.

7.3.2 Kelvin cell model Kraynik and Warren (1994) suggested that the Kelvin foam would be a good candidate for analysis. Figure 7.6a shows a complete cell of an ‘initial’ Kelvin foam, with edge cross-sections that are almost equilateral triangles.

7.4

Elastic moduli of the Kelvin foam 7.4.1 Uniform edge cross-sections When the foam strain remains below 1%, there is insufficient change in the foam geometry to cause non-linearity in the stress–strain relation. Hence, Young’s modulus characterises the foam compressive response. Dement’ev and Tarakanov (1970) calculated the Young’s modulus of the Kelvin foam for the [001] compression direction. Chains of edges, surrounding square faces connected corner to corner, take the compressive load (Fig. 7.6b) as they deform by bending. They showed, for low relative densities, that Young’s modulus E100 is given by 4

E100 ⫽

2EP

(b L)

2⫹b L

(7.8)


157

F

3 2

1

F (a)

Figure 7.6

(b)

(a) Kelvin foam cell with almost equilateral triangle edges (from Surface Evolver). (b) Dement’ev and Tarakanov’s model of the load path for [001] compression.

where EP is the polymer Young’s modulus, b the breadth of the edges with a square cross section, and L their length. The denominator contains a correction for the vertex volume. Equation (7.8) predicts, when R is small, that the normalised Young’s modulus E*001 ⫽ 0.63. Zhu et al. (1997a) analysed the small strain deformation of the Kelvin foam in the [001] and [111] directions using energy methods. In parallel, Warren and Kraynik (1997) analysed its general small strain deformation, using periodic boundary conditions on a small section of the lattice. They considered the deformation of a single vertex surrounded by four half-edges; this motif repeats throughout the lattice. They found the Kelvin foam to be slightly elastically anisotropic. Table – 7.1 gives the average Young’s modulus E of a polycrystalline, and hence isotropic, Kelvin foam, and shows how the assumed edge cross– section affects the constant in the equation between E and R2. The relative density R of a Kelvin foam is related to the (constant) edge cross-section area A and length L by equation (1.10), if the vertex volume is ignored. There are four slanting edges and eight horizontal half-edges in the structural cell of Figure 7.7, which has volume 4冑苳 2L2. The edges are treated as slender beams, whose curvature

158 Polymer Foams Handbook P P

O P

D

P

Figure 7.7

RUC for the [001] compression with boundary mirror planes. External forces P act in the z direction.

(the reciprocal of the radius of curvature R) is given by 1 d2 y ≅ R dx 2

(7.9)

where y is the deflection and x the position co-ordinate along the beam. This equation can be integrated twice to give the bending deflection. The foam Young’s modulus E100 is related to the half-edge stretching and bending compliances (Table 7.2) by 1 E100

⫽

1 ⎛⎜ L4 12L2 ⎞⎟⎟ L + ⎜⎜ (CB + 4CS ) ⎟⎟ = EA ⎠ 6 2 ⎝⎜ EI 2 2

(7.10)

The term L2/EA varies with R⫺1, whereas the term L4/EI varies with R⫺2, so for low R, it can be neglected. Substituting the I value for a Plateau border cross-section from Table 7.1, the foam Young’s modulus is E100 =

1.009ER2 1 + 1.514R

(7.11)

This confirms the values in Table 7.1, since E100 is only a few percent– ages larger than E. If equation (7.11) is plotted as a function of R,


159

the exponent in a power law fit is 1.95. The denominator in the equation contains a correction for edge compression. The foam elastic anisotropy can be expressed using Zener’s anisotropy factor A* and is defined by A∗ ⫽

2(1 ⫹ ν12 )G12 E100

(7.12)

where G12 is the shear modulus when shear stresses act on the [010] and [001] planes of the BCC lattice. In the limit of low R, the anisotropy factor for the Kelvin foam is A* ⫽

3 ⎛⎜ 5EI ⫹ GJ ⎞⎟ ⎟ ⎜ 2 ⎜⎝ 8EI ⫹ GJ ⎟⎟⎠

(7.13)

For Plateau border edge cross-sections, A* ⫽ 0.951, which is close to the value of unity for isotropic materials. This near isotropy is one reason for preferring the Kelvin model over other lattice models for open-cell foams. Gong et al. (2005a) considered the effects of edge shear in this structure, but these are relatively minor. The next level of approximation (Fig. 7.1) is to consider the complete edge and vertex geometry.

7.4.2 Non-uniform edge cross-sections The effects of the non-uniformity of edge cross-sections on the foam normalised Young’s modulus have been considered by two methods. Gong et al. (2005b) measured the edge width profiles of a number of edges in a PU foam, and used a polynomial fit function to describe the relation* ship. They predicted, using beam element FEA model, that E001 ⬵1.7. Mills (2007) used Surface Evolver software to create the foam geometry (see Fig. 1.3) and then solid-element FEA to analyse the stresses (details provided in Section 7.6). E*001 was predicted to be in the range 2.1–2.2 for foams of relative densities from 0.027 to 0.110. The edge bending moment M is largest close to the vertices, where the edge second moment of area I is largest, while it is zero at the mid-edge where I is lowest. Hence there is a lower edge curvature in the wet Kelvin foam than in the constant edge section model with the same R. The high value of E*001 illustrates the mechanical efficiency of this foam compared with a uniform edge section model. However, for compression in the [111] direction, E*111 is in the range 0.89–1.09 for R in the range 0.007–0.062; a significant amount of the foam compression is due to edge torsion, and there is a more uniform torque distribution along the edges. Consequently the wet Kelvin model is predicted to have a moderate elastic anisotropy.


7.5

Compression of the Kelvin foam with uniform edges 7.5.1 History of modelling Dement’ev and Tarakanov (1970) analysed the large strain deformation of a Kelvin foam under hydrostatic compressive loading and obtained solutions containing elliptic integrals. Warren and Kraynik (1997) used FEA to consider the general deformation of the unit cell of a Kelvin foam. The shapes of cell edges (of constant cross-section) were shown, but details of the deformation mechanisms were not spelt out. Zhu et al. (1997b) considered two compression directions, describing the deformation mechanisms. The [001] response is set out in the following section, while the [111] response has been superseded by FEA and is set out in Section 6.2.

7.5.2 Stress–strain response When a uniaxial stress σz is applied in the [001] direction (Fig. 7.7), the boundary planes of the structural cell remain mirror symmetry planes at high deformations, so the horizontal edges do not bend. The Elastica approach (Timoshenko and Gere, 1961) for the high elastic deflections of beams (edges) ignores axial and shear deformation compared with bending deformation. Axial edge deformation is minor for foams with R ⬍ 0.1, except when the foam is subjected to high tensile strains. The deformation symmetry means that only the half-edge OD need be analysed. A curvilinear co-ordinate s, with origin O, defines position in OD (Fig. 7.8), which lies in the yz plane. θ is the angle between the tangent to the edge and the z-axis. As the moment at O is zero and a constant force P acts on the edge, the moment at a general position is ⫺Py. The angle β at D is 45° for isotropic cells, but smaller for a model that represents PU foams that have risen. The differential equation for the beam curvature is

EI

dθ ⫽ M ⫽ ⫺Py ds

(7.14)

Differentiating this with respect to s, and using the relation dy ⫽ sin θ gives relationship ds

EI

d 2θ ⫽ ⫺P sin θ ds 2

(7.15)


161

P

O s

z

θ

zL

D β

Figure 7.8

yL

Force components acting on half-edge DO from Figure 7.7.The bending moment is zero at O.The projected edge lengths are y(α) and z(α) (Zhu et al., 1997).

Multiplying both sides of the equation by d␪/ds and integrating, using the boundary conditions at O such that the curvature is zero and θ ⫽ α, lead to 2

EI ⎛⎜ dθ ⎞⎟ ⎜ ⎟ ⫽ P(cos θ ⫺ cos α) 2 ⎜⎝ ds ⎟⎟⎠

(7.16)

P is introduced as an abbreviation for sin α/2 and φ is defined by sin

θ ⫽ p sin φ 2

(7.17)

After some algebra, the compressive engineering stress is shown to be ⫺σz ⫽

2EIF 2 (α) L4

(7.18)

where EI is the edge bending stiffness and F(α) is an elliptical integral defined by F(α) ≡ ∫

π 2 δ

dφ 1 ⫺ p2 sin2 φ

(7.19)

162 Polymer Foams Handbook This is a function of the inclination α of the edge midpoint O, while the lower limit δ of integration depends on the edge inclination β at D through δ ⫽ sin⫺1 ( sin β /2 sin α/2 )

(7.20)

For isotropic-shaped cells, β ⫽ 45° is substituted into this equation. For Plateau border edge cross-sections, ignoring the material in the vertices, equation (7.18) becomes σ2 ⫽ 0.2379ER2F2(α)

(7.21)

The compressive strain is obtained from the vertical projection z(α) of the half-edge. As the undeformed half-edge has a height L/2冑苳 2, the foam tensile strain is

εz ⫽

z(α) ⫺ L 2 2 L 2 2

⫽

2 2 z(α) ⫺ 1 L

(7.22)

As the strain tends to zero, the slope of the stress–strain curve becomes Young’s modulus E100. At a strain of 0.1%, the secant Young’s modulus is E100 ⫽

σz εz

⫽ 1.007 ER2

(7.23)

effectively the same as equation (7.11), derived using small deflection bending theory. Figure 7.9 shows the predicted shapes of the half-edge OD for various foam strains. We see in the next section that long-range buckling occurs at foam strains ⬎0.2, so the shapes for higher strains are only of academic interest.

7.5.3 Long-range buckling Laroussi et al. (2002) showed that multi-cell buckling modes, for [001] direction compression of the Kelvin foam, cause a non-linear response; however, they did not predict the shape of the compressive stress–strain curve. Gong and Kyriakides (2005) considered RUC


163

0.8 0 0.15

0.6

z

0.3 0.4

0.45 0.6

0.2

0

0

0.2

0.6

0.4

0.8

1

y

Figure 7.9

Predicted shapes of the edge DC for [001] compression and for foam compressive strains indicated (Zhu et al., 1997).

containing 2–20 whole Kelvin cells in the [001] direction, one cell wide and one cell thick, with periodic boundary conditions. In general the cells were elongated in the rise direction, but they could be made isotropic. Using FEA with beam elements to represent Plateau border edges with non-uniform cross-sections, they predicted for a polyester urethane foam (R ⫽ 0.025, EP ⫽ 69 MPa, ν ⫽ 0.49) that collapse occurs by buckling when the normalised stress ⬵ 0.15 at a foam compressive strain ⬵ 0.2. Figure 7.10 shows the buckling mode. At higher strains, the compressive stress was predicted to be nearly constant. Gong et al. (2005b) used a Bloch wave method to confirm the onset of instability in the Kelvin foam model. Hence, the small RUC used in the last section is inappropriate for high-strain compression in the [001] direction. Nevertheless, Section 7.6.2 shows that for [111] direction compression, a small RUC appears to be adequate.

7.6

FEA model of wet Kelvin foam 7.6.1 [001] direction compression Mills (2006) used a triangular prism, with 90° and 45° angles, as the RUC for [001] direction compression; this is a quarter of the structure cell shown in Figure 7.7. As described in Chapter 1, Surface Evolver


3

3 2

Figure 7.10

1

1

2

Predicted buckling of larger RUCs for the Kelvin foam compressed in the [001] direction (Laroussi et al., 2002 © Elsevier).

software was used to simulate the foam geometry. Figure 7.11a shows the edge shape, for a wet foam with R ⫽ 0.062, at a compressive strain of 18%; three waves have developed along the cusp that borders the hexagonal face. This local buckling, not considered in the Zhu et al. (1997b) theory, has a negligible effect on the compressive stress–strain relationship, which is only slightly non-linear (Fig. 7.11b). This model is not valid above this strain level, since long-range buckling, discussed in the last section, comes into play at higher strains.

7.6.2 [111] direction compression A set of axes are chosen, with the z-axis parallel to the lattice [111] direction. The RUC is a prism with an equilateral triangular crosssection, having at its centre a threefold screw axis (Fig. 7.12). The sides of the prism are mirror symmetry planes, while there are periodic boundary conditions between the top and base of RUC. The symmetry means that: (a) no tensile and shear forces act on the ends of initially horizontal (meaning in the xy plane) edges. They deform into circular arcs, lying in a vertical plane. (b) The helix of edges does not rotate relative to the prism.

Chapter 7 Micromechanics of open-cell foams 13

165

0.35

4

Normalised stress

13 10 7 4 4 4 3

0.3 0.25 0.2 0.15 0.1 0.05

1 2

0 (a)

0


0.2

(b)

Figure 7.11

Wet Kelvin foam with R ⫽ 0.062: (a) principal tensile stress contours (MPa), for 18% compressive strain in the [001] direction and (b) compressive stress–strain response (Mills, 2007).

A1 A1 C2

2 1

Bounding planes A2

Section of edge helix

A2

C2 E1

E2 C1

C1

B1 Cut end face B2

(a)

Figure 7.12

B1

Periodic between B2

(b)

RUC for [111] direction compression for edge width S ⫽ 0.2: (a) end view and (b) side view.The cut edges at the sides are labelled A1 to C2 and at the ends E1 and E2 (Mills, 2007).

The force P on the helix of edges is related to the applied stress by P⫽

4 3

L2σz

(7.24)

FEA of this RUC (Mills, 2006) largely confirmed the earlier analysis (Zhu et al., 1997b) using Euler–Bernoulli beams, in which the effects of shear were ignored. However it revealed errors in earlier predictions of the foam shape at strains ⬎50%. Figure 7.13 shows that the normalised stress σ* exhibits a maximum in the range 0.09–0.11 (depending on the foam relative density),

166 Polymer Foams Handbook 0.12 0.3 0.1

Normalised stress

0.1

PL

0.2

PL

0.08 0.06 0.04 0.02 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Compressive strain

Figure 7.13

Compressive normalised stress–strain responses of the wet Kelvin foam in the [111] direction labelled with S for elastic and PL for elastic–plastic materials (Mills, 2007).

Figure 7.14

Shape of one edge of wet Kelvin foam with R ⫽ 0.009 for [111] compressive strains (left to right) 0%, 20%, and 40%.The contours of von Mises stress are from 0 to 8 MPa.

when the compressive strain is 30%, with a plateau to about 60% strain. There is no sudden change of slope, as in Figure 7.5c, because the edges do not buckle at any stage in the deformation. Figure 7.14 shows a single complete edge in a Kelvin foam of R ⫽ 0.009 under [111] compression, with the compression direction horizontal. The low relative density allows the edge shape to be better seen. As the compressive strain increases, the edge rotates away from the compression axis, twists, and bends. The diameter of the ‘hole’ along the axis of the helix enlarges when the foam is deformed. This increases the torque on the vertices for a given axial load on the RUC but reduces the foam secant modulus. The prism of the edges


167

0.08

0.06

Lateral strain

0.04

0.02

0.3 0.1

0 0.2 −0.02 −0.04 −0.06

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Compressive strain

Figure 7.15

Predicted lateral strain vs. [111] compressive strain for wet Kelvin foams with S values shown (Mills, 2006).

expands laterally to a maximum of about 10% when the compressive strain is 20% and then decreases (Fig. 7.15). Hence the overall Poisson ratio of the model is very low for large compressive strains.

7.7

Irregular foam models As discussed in Section 1.6.2, there is no single way to simulate the structure of PU foams. The number of cells considered in a model is limited because of computer restrictions. Most researchers assume rules for cell generation that include random processes and then generate many examples of a model. The predicted mechanical properties vary from example to example, so averages must be taken. Van der Burg et al. (1997) randomised foams based on BCC and FCC lattices. For cell nuclei on a BCC lattice, a Voronoi tesselation creates the Kelvin closed-cell foam; removing the faces leaves an open-cell foam. To introduce randomness, they moved the nuclei in random directions from the BCC positions. The nuclei displacements δ were chosen from a uniform distribution in the range [0 to δmax], where δmax could be up to 75% of the lattice point separation. However the displacement vectors should have been uniformly distributed in a sphere of radius δmax. Their model was a cube containing 6 by


Reduced stress σ

−

α = 0.7 0.15

0.10

0.05

0.00 0.0 (a)

Figure 7.16

(b)

Predicted Simulated

0.1


0.5

0.6

(a) View of the undeformed foam structure and (b) reduced stress vs. compressive strain. α is the regularity parameter, while the ‘predicted’ relationship is a curve fit of the data (Zhu and Windle, 2002 © Elsevier).

6 by 6 cells. There tended to be chains of edges nearly aligned with the original [001] directions of the lattice, so the model may be stiffer in these directions than in the [111] directions. They predicted, for edges with circular cross-sections, that Young’s modulus rises by 46% as randomness is increased. Intuitively this seems improbable. Zhu et al. (1999) used a Voronoi tessellation of 27 nuclei placed at random positions in a cube. The geometry of such models differs from real PU foams (Chapter 1). The total edge length increased with randomisation, meaning that the edge cross-sectional area decreased (at a given R value). Hence, it is not easy to separate the effects of edge direction randomisation and the edge length-to-breadth ratio. Using periodic boundary conditions and uniform edge cross-sections, they predicted Young’s moduli for irregular structures that were about 50% higher than a regular Kelvin foam. Zhu and Windle (2002) used the same model (Fig. 7.16a) to predict a compressive stress–strain graph (Fig. 7.16b) that is more non-linear than that for the Kelvin model. The FEA modelling gave better predictions of lateral strains than the Kelvin model. However, Gan et al. (2005), who made similar computations for edges with circular cross-sections, predicted no increase in Young’s modulus with foam irregularity. Since the cell generation rules affect the predicted mechanical properties, there is uncertainty about the effects of foam irregularity. The annealed Voronoi models generated by Kraynik et al. (2003) have geometries closer to real foams than the initial Voronoi models. However, as discussed in Chapter 1, some details still differ from polymer foams, and the foams are ‘monodisperse’ (the spheres, initially packed into the RUC, had the same volume). Elliott et al. (2002) proposed using the vertex positions measured in real foams,


169

6

Stress (MPa)

5 4 3 2

Frictionless

1

No sliding Experimental

0

0

0.02

0.04

0.06

0.08

0.1

Strain

Figure 7.17

Predicted compressive stress–strain curve for rigid PU foam of relative density 0.33, compared with experiment (Youssef et al., 2005 © Elsevier).

with the edges modelled as beams, to construct FEA models. However, the forces and moments on the cut edges at the boundary of the sample are unknown. Also, it is not clear how to alter the structure to achieve cyclic boundary conditions. Maire et al. (2003) also discuss the difficulty of constructing FEA models from X-ray tomography data, and whether to use beam or 3D elements. Then Youssef et al. (2005) carried out a 3D element FEA for a rigid PU foam of relative density R ⫽ 0.33. This contained spherical bubbles, only some of which touch or overlap; so the microstructure (Fig. 1.2a) differs from those emphasised in this chapter. The predicted compressive stress strain curve, for a polymer Young’s modulus of 1.6 GPa, Poisson’s ratio of 0.38, initial yield stress of 40 MPa, and hardening coefficient 1.25 MPa, is close to the experimental data (Fig. 7.17). Knackstedt et al. (2006) used a similar approach for rigid PU foams, of a range of relative densities from 0.29 to 0.56, converting X-ray tomograph images into finite element models. To overcome the boundary condition problem, they surrounded the sampled 2003 voxels (or elements) by a 15 voxel thick boundary phase and then adjusted the modulus of the boundary phase until there was a convergence between this and the average modulus of the whole model. They predicted Young’s modulus, bulk modulus, and Poisson’s ratio as a function of foam relative density. Their predictions were close to the relationship given by equation (7.3) with the constant C1 ⫽ 1.0. However, no experimental modulus measurements were made.

7.8

Anisotropic cell shapes Most slabstock PU foams have anisotropic cell shapes. Zhu et al. (1997b) made a Kelvin foam with uniform edge cross-sections


Normalised stress (σ/Erise)

0.08

Rise

0.06

0.04 Transverse 0.02

0

0

0.1

0.2

0.3

0.4

Compressive strain (%)

Figure 7.18

Predicted stress normalised with the rise direction Young’s modulus vs. compressive strain for anisotropic Kelvin foams compressed in (001) directions (redrawn from Gong et al., 2005).

anisotropic, with cells elongated along a [111] axes. The angle γ0, between ‘sloping’ edges and the compressive stress axis, reduces from 35.2° for equiaxed cells to 28° in a cell with a height to width ratio of 1.26. The slanting cell edges become 18% longer than the horizontal edges. The predicted dimensionless stress–strain curve showed a greater non-linearity compared with the equiaxed result, with the stress approaching a plateau for strains ⬎40%. Gong et al. (2005b) considered the compression of large RUCs of Kelvin foam cells with an anisotropy ratio of 1.25, having edges with a varying cross-section. The stress–strain curves were predicted for compression in the rise direction and a transverse direction, both of which were (001) type directions in the lattice. The models were given initial imperfections of 0.01 in the traverse direction and 0.25 in the rise direction. The predicted Young’s modulus in the transverse direction, for a relative density of 0.025, was only 47.5% of that in the rise direction. Their normalised stress–strain curves are compared on the same figure (Fig. 7.18). Many of the features of the experimental stress–strain curves were predicted.

7.9

Non-linear polymer response Mills (2006) made an elastic–plastic FEA for [111] direction compression of wet Kelvin foams. The elastic–plastic response of the PU


171

was taken from experimental data on single cell edges (van der Heide et al., 1999), with an initial yield stress 10 MPa, increasing to 31.5 MPa at a strain of 2.5 (Fig. 2.9). Figure 7.13 shows that the foam compressive response is only affected by plasticity for foam strains greater than 0.3 (for R ⫽ 0.026). Although small regions of the bent and twisted cell edges yielded, most of the PU remained elastic. Consequently, the non-linearity of the foam compressive response is mainly due to the geometry changes in the foam. The elastic material constants can also be replaced by the appropriate viscoelastic polymer properties to predict the viscoelastic foam behaviour (Chapter 19).

7.10

Strain localisation Lakes et al. (1993) argued that strain localisation is possible in compressed open-cell PU foams, without a maximum in the compressive stress–strain curve, since both the foam density and the edge orientation vary from point to point. It is commonly observed that one cell collapses before its neighbours, but this has little effect on the stress–strain curve. In honeycomb structures, with a uniform cell size, compressed in a direction perpendicular to the cell axes, a layer of highly compressed cells forms perpendicular to the stress axis. This band then thickens as the overall mean strain increases. Such a band must fit together with the less deformed material above and below, so it is constrained to have zero lateral strain. In the wet Kelvin foam model, with a uniform cell size, a similar behaviour is possible. When the model with material plasticity is compressed in the [111] direction, a maximum in the compressive stress is predicted. At higher strains, edge touching will cause hardening. As there are two possible strains for the same stress, the compressive strain could become inhomogeneous. As a band of compressed cells thickens, the macroscopic compressive stress remains constant, while the mean strain in the foam increases. Hence there is a plateau in the stress vs. mean strain curve. However, in PU foams with a range of cell sizes and orientations, such behaviour has rarely been observed.

7.11

Modelling edge touching FEA modelling of edge touching in the Gibson and Ashby model predicted a sudden upturn in the compressive stress–strain graph at a compressive strain of 50% (Fig. 7.5b). The [111] direction compression of the wet Kelvin foam, shown in Figure 7.14, was repeated for a relative density R ⫽ 0.028. The thicker edges were predicted to touch at a compressive strain of 66%. However, since edge-to-edge contact could not be simulated, high-strain hardening could not be

172 Polymer Foams Handbook predicted. Although modelling of edge contact should soon be possible, the predictions are likely to differ from the response of PU foams containing a range of cell sizes. Edge-to-edge contact should occur over a wide range of mean compressive strains for such a foam, compared with a uniform cell-size model; so hardening should be less sudden than that predicted in Figure 7.5b. In beam element models of the Kelvin foam, the edges have negligible width. Gong and Kyriakides (2005) used a contact spring approximation to limit the approach of vertices which are aligned along the [001] compression direction. Although it predicted an upturn in the stress–strain curve at high strains, this empirical model cannot consider the geometric interactions between Plateau bordershaped edges.

7.12

Comparison with experiment The normalised Young’s modulus E* predicted by FEA for the wet Kelvin foam loaded in the [111] direction is 1.0 ⫾ 0.1. This confirms the order-of-magnitude estimate of Gibson and Ashby (1988) who fitted a quadratic scaling law to experimental data for reduced Young’s moduli (foam modulus/polymer modulus) vs. density. Their data are very scattered, the possible reasons being: (a) Uncertainty in estimating the polymer Young’s modulus. The mechanical properties of bulk PU cannot be used, because this will not have the same chemical structure. The properties of the PU in foam usually change with the foam density (see Chapter 2) due to changes in the polyurea hard segment content. Most PU foams cannot be remoulded into a solid for mechanical property measurement. (b) PU foams tend to have anisotropic cells, so there are different moduli in the rise and perpendicular directions. (c) Polymer viscoelasticity causes the stress to rise above the elastic prediction while the strain is rising, but to fall below when the strain is decreasing. The Kelvin model predictions are consistent with the average of experimental data. Thus Young’s modulus of open-cell foams can be related to their relative density and the polymer modulus, to within a factor of two. This is reasonable given the uncertainties in material properties and the foam microstructure. The compressive stress and lateral strain of a PU foam were measured for compression along the foam-rise direction at a nominal strain rate of 3 ⫻ 10⫺3 s⫺1. The stress–strain relationship is initially linear with a Young’s modulus of 14.7 kPa (Fig. 7.19). For compressive strains less than 12%, there is a linear relationship between the


173

6 5

PU rise Wet Kelvin

Stress (kPa)

4 PU in plane

3 2 1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Compressive strain

Figure 7.19

Predicted stress–strain curve for a wet Kelvin foam compressed in the [111] direction for an elastic material compared with data for a PU foam of density 31 kg m⫺3.

lateral and compressive strains, with a Poisson’s ratio of 0.09. At higher strains, the lateral strain has a plateau value, suggesting a change in deformation mechanism. The wet Kelvin model for a relative density of 0.025 predicted (Fig. 7.15) a zero lateral strain at foam compressive strains of 60%, but the intermediate predictions differ from experimental data. There is a stress peak at 10% strain for compression along the foam-rise direction, while there is a monotonic stress increase for compression in the plane of the foam sheet. The experimental curve for the foam-rise direction lies above the prediction, whereas the in-plane compression data are close to the prediction up to 40% strain. At 70% compressive strain the experimental curves rise rapidly as edges touch. Comparison with Figure 7.18 shows that an anisotropic model is required to explain the foam response. The contribution of viscoelasticity to the compressive response of PU foams is considered in Chapter 19.

References Dement’ev A.G. & Tarakanov O.G. (1970) Effect of cellular structure on the mechanical properties of plastic foams, Polym. Mech. 6, 519–525. Elliott J.A., Windle A.H. et al. (2002) In-situ deformation of an open-cell flexible polyurethane foam characterised by 3D computed tomography, J. Mater. Sci. 37, 1547–1555.

174 Polymer Foams Handbook Gan Y.X., Chen C. & Shen Y.P. (2005) Three-dimensional modelling of the mechanical property of linearly elastic open cell foams, Int. J. Solid. Struct. 42, 6628–6642. Gibson L.J. & Ashby M.F. (1988) Cellular Solids, Pergamon, Oxford. Gong L. & Kyriakides S. (2005) Compressive response of open-cell foams, Part 2. Initiation and evolution of crushing, Int. J. Solid. Struct. 42, 1381–1399. Gong L., Kyriakides S. & Jang W.Y. (2005a) Compressive response of open-cell foams, Part 1. Morphology and elastic properties, Int. J. Solid. Struct. 42, 1355–1379. Gong L., Kyriakides S. & Triantafyllidis N. (2005b) On the stability of Kelvin cell foams under compressive loads, J. Mech. Phys. Solid. 53, 771–794. Hamza R., Zhang X.D. et al. (1997) Imaging open-cell polyurethane foams via confocal microscopy, in Polymeric Foams, Ed. Khemani K., ACS 669, American Chemical Society, Washington DC. pp. 165–177. Hilyard N.C., Ed. (1982) Mechanics of Cellular Plastics, Macmillan, New York. Knackstedt M.A., Arns C.H. et al. (2006) Elastic and transport properties of cellular solids derived from three-dimensional tomographic images, Proc. Roy. Soc. London A 462, 2833–2862. Kraynik A.D., Reinelt D.A. & van Swol F. (2003) Structure of monodisperse foam, Phys. Rev. E 67, 031403. Kraynik A.M. & Warren W.E. (1994) In Low Density Cellular Plastics, Eds. Hilyard N.C. & Cunningham A., Chapman and Hall, London. Lakes R., Rosakis P. & Ruina A. (1993) Microbuckling instability in elastomeric cellular solids, J. Mater. Sci. 28, 4667–4672. Laroussi M., Sab K. & Alaoui A. (2002) Foam mechanics: nonlinear response of an elastic 3D-periodic microstructure, Int. J. Solid. Struct. 39, 3599–3623. Maire E., Fazekas A. et al. (2003) X-ray tomography applied to the characterization of cellular materials, related finite element modelling problems, Compos. Sci. Tech. 63, 2431–2443. Mills N.J. (2005) Plastics: Microstructure and Engineering Applications, 3rd edn., Butterworth Heinemann, London, p. 288. Mills N.J. (2006) Mechanics of the wet Kelvin foam in compression, Int. J. Solid. Struct. (accepted). Mills N.J. (2007) The high strain mechanical response of the wet Kelvin model of open cell foams, Int. J. Solids Struct. 44, 51–65. Mills N.J. & Gilchrist A. (2000) The high strain extension of opencell foams, J. Eng. Mat. Tech. ASME 122, 67–73. Phelan R., Weaire D. et al. (1996) The conductivity of a foam, J. Phys. Condens. Matter 8, L475–L482. Renz R. & Ehrenstein G.W. (1982) Calculation of deformation of cellular plastics by the finite element method, Cell. Polym. 1, 5–13.


175

Shulmeister V. (1998) Modelling of the mechanical properties of lowdensity foams, Ph.D thesis, Technical University of Delft, Shaker Publishing, Maastricht. Timoshenko S.P. & Gere J.M. (1961) Theory of Elastic Stability, McGraw-Hill, New York. van der Burg M.W.D., Shulmeister V. et al. (1997) On the linear elastic properties of regular and random open-cell foams, J. Cell. Plast. 33, 31–54. Van der Heide E., van Asselen O.L.J. et al. (1999) Tensile deformation behaviour of the polymer phase of flexible polyurethane foams and polyurethane elastomers, Macromol. Symp. 147, 127–137. Warren W.E. & Kraynik A.M. (1997) Linear elastic behaviour of a low-density Kelvin foam with open-cells, ASME J. Appl. Mech. 64, 787–794. Warren W.E., Neilsen M.K. & Kraynik A.M. (1997) Torsional rigidity of a Plateau border, Mech. Res. Commun. 24, 667–672. Youssef S., Maire E. & Gaertner R. (2005) Finite element modelling of the actual structure of cellular materials determined by X-ray tomography, Acta Mater. 53, 719–730. Zhu H.X. & Mills N.J. (1999) Modelling the creep of open-cell polymer foams, J. Mech. Phys. Solid. 47, 1437–1457. Zhu H.X. & Windle A.H. (2002) Effects of cell irregularity on the high strain compression of open-cell foams, Acta Mater. 50, 1041–1052. Zhu H.X., Knott J.F. & Mills N.J. (1997a) Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells, J. Mech. Phys. Solid. 45, 319–343. Zhu H.X., Mills N.J. & Knott J.F. (1997b) Analysis of high strain compression of open-cell foams having tetrakaidecahedral cells, J. Mech. Phys. Solid. 45, 1875–1904. Zhu H.X, Hobdell J.R. & Windle A.H. (1999) The effects of cell irregularity on the elastic properties of open-cell foams, Acta Mater. 48, 4893–4900.

Chapter 8

Air flow in open-cell foams

Chapter contents 8.1 Introduction 8.2 Air-flow measurement 8.3 Models for air-flow resistance 8.4 Air flow during foam impact compression 8.5 Sound absorption in foams 8.6 Filters References

178 178 184 192 198 200 201


8.1

Introduction Air-flow properties are important when open-cell foams are used as air filters or for sound absorption. Sound consists of pressure waves in air in the frequency range 50 Hz to 10 kHz. Metal open-cell foams, based on polyurethane (PU) foams, are used for heat exchangers. Airflow measurements are also used to characterise the open-cell content of foams; for example Jones and Fesman (1965) used the variation of air-flow values with position in a large bun of PU foam to characterise the product inhomogeneity. Zhang et al. (1997) showed that silicone surfactants control face opening in PU foams, affecting the air-flow values. The air-flow resistance of foams and its modelling in terms of microstructural parameters (cell size, the proportion of open-cell faces, and foam density) will be dealt with first. Cell size strongly influences air-flow properties, in contrast with its minor effect on mechanical properties. The compressive impact resistance of some large open-cell foam products may be influenced by air flow, so the appropriate experimental data is described and the phenomena modelled. Finally, the links between sound absorption in foams and air flow will be explored.

8.2

Air-flow measurement 8.2.1 Equipment To measure the air-flow resistance of flexible foams, BS 4443-6: 1991 describes a metre long, thick-walled, steel tube of internal diameter 75 mm, with air supplied to one end, the sample in the middle, and orifice-type gas flow meters at the other end. Typical samples are – 50 mm cubes. The mean air velocity V through the foam is related 3 1 to the steady state gas flow Q m s and the foam cross-sectional area A by V

Q A

(8.1)

Mills and Lyn (2002) fitted an electronic differential pressure transducer, with a 35 kPa range, between the chambers on either side of the sample. The pressure gradient through the foam is calculated from the differential pressure ∆p and foam length L as ∆p/L. The largest pressure gradients used did not compress the foam in the direction of air flow. It was not necessary to seal the surfaces of the foam to the sample holder, such sealing had negligible effect on the measured pressure gradient.

Chapter 8 Air flow in open-cell foams

179

8.2.2 Data treatment Both Gent and Rusch (1966), and Hilyard and Collier (1987) related the pressure drop ∆p across a foam length L to the air velocity V using p η ρ V V2 L K B

(8.2)

where η is the air viscosity 18 106 Pa s, and ρ its density 1.29 kg m3. The foam permeability K (m2) and inertial flow coefficient B (m) are material constants. Acoustic engineers, such as Cummings and Beadle (1993), prefer to use the equation P σV τV 2 L

(8.3)

where the air-flow resistivity σ and inertial flow resistivity τ combine the properties of the fluid and the porous medium. Air-flow resistivity has units rayl m1, where the rayl is the derived SI unit for acoustic impedance. Hilyard and Collier (1987) showed that B and K, for several types of open-cell PU foam, were functions of the applied compressive strain. They attributed the first term on the right-hand side of equation (8.2) to laminar air flow and the second to turbulent air flow. Gent and Rusch (1966) commented that turbulent air flow should only occur when the Reynolds number Re 2000, that is at much higher air velocities than those used experimentally. If the flow velocity is 5 m s1 and the cell diameter is 1 mm, Re 350. Observing that the pressure gradient depended on the square of the air velocity, they suggested that turbulent flow occurred for Re 艑 1. However, there have been no observations of turbulent flow in such foams. Ruth and Ma (1992) suggested that changes of air-flow direction in porous materials cause secondary flows, and that these, rather than turbulence, cause the V2 term.

8.2.3 Data for PU foams with fully open cells Cummings and Beadle (1993) measured the variation of σ and τ with mean cell diameter, for a range of Recticel reticulated PU foams, of densities in the range 23–32 kg m3. They used the inverse of the pores per mm value (from a linear intercept count) as a mean cell diameter D. Their results, expressed in terms of B and K, are plotted on logarithmic scales vs. D2 and D, respectively (Fig. 8.1). The linear-fit slopes, of 1.1 and 1.2 respectively, are sufficiently close to 1.0 to suggest that

180 Polymer Foams Handbook 4 3.5

3

log (η/B)

log (η/K)

3.5

2.5

3

2 2.5 1.5 −3

−2.5

−2 log

(a)

Figure 8.1

−1.5

−1

−0.5

(η/D 2)

1

1.5

2

log (η/D)

(b)

Variation of B and K with mean cell diameter (recalculated from Cummings and Beadle, 1993 © Academic Press).

Permeability (10−9 m2)

20

10 9 8 7 6 5 4 3

2 0.2

0.3

0.4

0.5 0.6 0.7 0.8 0.9 1

2

Mean cell diameter (mm)

Figure 8.2

Air permeability of PU foams (Hilyard and Collier, 1987 © RAPRA).

K D2

and

BD

(8.4)

Hilyard and Collier (1987) found that, for a range of PU foams from three UK manufacturers, K varied with D2.08 (Fig. 8.2), that is it could be fitted by equation (8.4). There has been no systematic study of the effect of PU foam density, at a constant cell size, on air-flow parameters. Bhattacharya et al. (2002) investigated the air-flow parameters of open-cell aluminium


181

80

Pressure gradient (kPa/m)

70 60 50 40 30 20 10 0

Figure 8.3

0

1

2 Air velocity (m/s)

3

Pressure gradient vs. air-flow velocity for steady state air flow through uncompressed PU chip foam of length 50 mm, fitted with a quadratic function (Mills and Lyn, 2002).

foams based on PU foams, with relative densities in the range 0.02– 0.11. At a mean cell diameter of 2 mm, K 艑 50 109 m2 and at 4 mm, K 艑 200 109 m2, so the data roughly falls on the extrapolated trend line in Figure 8.2. Paek et al. (2000) studied similar foams for air velocities up to 3.7 m s1, and found pressure gradient vs. air velocity relations of similar shape to Figure 8.3.

8.2.4 Data for compressed PU foams with fully open cells Hilyard and Collier (1987) compressed foam samples, in a direction perpendicular to the air-flow direction, to strains ε between 0% and 70%, to fit a 50 mm cube sample holder. They proposed semi-empirical equations for the variation of B and K with the strain 3 ⎪⎧ 1 ε R ⎪⎫⎪ B B0 ⎪⎨ ⎬ (1 ε) ⎪⎪⎩ (1 ε)(1 R) ⎪⎪⎭

(8.5)

3 ⎪⎧ 1 ε R ⎪⎫⎪ K K0 ⎪⎨ ⎬ (1 ε)2 ⎪⎪⎩ (1 ε)(1 R) ⎪⎪⎭

(8.6)

based on an approximate model for the permeability of isotropic porous solids. This assumed that the compressed foam was isotropic, with an


Air-flow parameters for velocity range 0.5–5 m s1 through chip foam of density 64 kg m3 Compressive strain (%) 0 20 40 60 70

Table 8.2

Permeability K (109 m2)

Air inertia coefficient B (µm)

4.76 4.04 4.58 3.13 2.48

153 109 73 75 58

Air-flow parameters for slow-recovery PU foams Foam Royal Medica Sunmate soft Sunmate medium Sunmate firm Pudgee

Permeability K (109 m2)

Air inertia B (µm)

1.65 0.50 0.84 0.26 0.77 0.15 1.04 0.42 0.60 0.19

100 78 560 480 260 370 650 620 73 8

increased density of obstructing objects. They showed that B and K, for several types of hot cure and high resiliency PU foams, varied with the applied compressive strain according to equations (8.5) and (8.6) to within the experimental error. Their maximum air velocity was 0.6 m s1. Mills and Lyn (2002) measured the air-flow resistance of rebonded PU foam, and fitted ∆p vs. V data (Fig. 8.3) with equation (8.2). The B and K values (Table 8.1) decreased linearly with increasing compressive strain. However it was not possible to determine whether a linear decrease, or equations (8.4) and (8.5), best fitted the data.

8.2.5 Data for PU foams with partly open cells Slow-recovery PU foams tend to have circular holes in the cell faces (Chapter 2, Section 6.4). Curiously, their air inertia coefficient B is negative (Table 8.2). This is not an artefact of microstructural damage at high air velocities, since the air-flow values are repeatable for a cycle of increasing and decreasing velocity tests. Figure 8.4 shows typical pressure gradient vs. flow velocity measurements for two foams. As the sample lengths are 50 mm, the maximum pressure gradients are approximately 10 kPa m1. Modelling will be used later to explain this effect. Figure 8.5 shows a positive correlation between K and the mean face hole area of the Royal Medica and Sunmate foams; the Pudgee foam, of a slightly different microstructure, has a significantly lower K value.


183

Differential pressure (kPa)

1.2

1.0

0.8

Pudgee SunMedium

0.6

0.4

0.2

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Air velocity (m s−1)

Figure 8.4

Pressure differential vs. air velocity for 2 makes of slow-recovery foam, fitted with a quadratic function (Fitzgerald et al., 2004).

6

Permeability K(10−9 m2)

5 4 Theory 3 2

Royal Medica

1 Pudgee 0

0

0.5

1

1.5

2

2.5

3

3.5

Hole area (104 mm2)

Figure 8.5

Permeability K vs. mean cell face hole area for PU foams, of cell diameter approximately 0.5 mm, compared with predictions of the circular face hole Kelvin foam model for a 0.6 mm cell diameter (Fitzgerald et al., 2004).


8.3

Models for air-flow resistance The Gent and Rusch (1966) model is considered first; although its geometry is simpler than that of PU foams, it is easy to visualise the axisymmetric flow field, and the effects of cell size. Separate models are then considered for the different microstructures of slabstock and slow-recovery PU foams.

8.3.1 Gent–Rusch model Gent and Rusch (1966) suggested that turbulent losses occur as air passes through a series of constrictions in the foam. Their model, for a foam with cell diameter DC, consists of many parallel straight tubes of diameter DC, with faces perpendicular to the axis at spacing DC, each containing a small central hole. The gaps between the tubes are filled with solid. Hence the model geometry is a simplification of the microstructure of PU foams. They used the theory of Kozeny (1927) for porous rocks, which is discussed critically by Scheidegger (1960), who noted that it ignored the influences of conical flow near the constrictions in the (collection of) variable cross-section flow channels. Thus, the Kozeny theory is inappropriate for foams with multiple branching flow channels, in which much of the flow resistance comes from restricted holes in faces. It gives the permeability, for channels with circular cross-sections, as K 0 .5

φ3 TS 2

(8.7)

This involves the porosity φ, the specific surface area S, and the tortuosity T (defined as the actual flow path divided by the direct flow path across the material). Substituting T 1 for straight tubes, Gent and Rusch predicted a permeability K

DC2 32

(8.8)

which is unaffected by the hole area. The predictions of equation (8.8) are wrong for low density foams, since the DC2 relation breaks down for the large cell diameters used in filters, while the hole area has a significant effect on permeability. Prior to the advent of computational fluid dynamics (CFD), they were unable to consider the velocity distribution across the pipe or the effect of one constriction on the pressure drop at the next constriction.


185

DC Pipe wall

Cell face

Periodic boundary

(a)

(b)

Figure 8.6

Periodic boundary

0.4

0.8 1.2 1.6

DH

Central axis

2 4 6 8

CFD of air flow from right to left in the Gent–Rusch model, with velocity vectors and contours (m s1) for a pressure gradient of 10 kPa m1, DH/DF 0.2, for cell diameters: (a) 0.5 mm and (b) 2.0 mm (Fitzgerald et al., 2004).

Fitzgerald et al. (2004) used a CFD programme (Fluent Inc, 2003) to analyse laminar air flow through the Gent–Rusch model geometry. The repeat unit is from the middle of one cell to the middle of the next cell (Fig. 8.6). The pressure gradient, applied to the repeating unit, was 10 kPa m1. Periodic velocity boundary conditions were applied between the left- and right-hand edge boundaries. The axisymmetric mesh had typically 2400 elements. For a cell diameter DC 1 mm and DH/DC 0.2, K 0.5 109 m2. This compares with the prediction of equation (8.8), for DC 1 mm, of K 31 109 m2, independent of the hole size. Air velocity maps reveal why the permeability varies in a non-linear way with cell diameter. For a 0.5 mm cell diameter (Fig. 8.6a), the streamlines diverge from the hole and there is only a high velocity region in a small central region downstream of the hole. However, for a 2 mm cell diameter, the flow streamlines are almost parallel after they exit the hole (Fig. 8.6b), and the airstream has sufficient momentum for the high velocity region to extend to the next orifice.

186 Polymer Foams Handbook 8.3.2 Gent–Rusch model with distorted faces The negative B value for slow-recovery foams suggests that a velocitydependent phenomenon affects the foam permeability. Cell face distortion is a possible explanation. CFD had shown that the pressure differential, from one side to the other side of a hole-containing face, was nearly uniform. ABAQUS Standard finite element analysis (FEA) was used to predict the shape of a cell face of relative hole diameter 0.3 (Fig. 8.7), with a uniform pressure of 10 Pa applied to one side. The face thickness at its outer circumference was 2% of the face diameter (i.e. 20 µm for a 1 mm diameter face), and its thickness tapered linearly to zero at the inner diameter. The PU was assumed to have a Young’s modulus 5 MPa and Poisson’s ratio 0.4. The maximum deflection, at the edge of the hole, was predicted to be 49 µm in the flow direction. Subsequent CFD for the Gent–Rusch model with distorted faces showed (Table 8.3) the effect of face distortion is small for laminar flow, but significant for turbulent flow. The latter computations used the standard k–ε model, where the default Fluent parameters are Cµ 0.09, C1ε 1.44, C2ε 1.92, and the TKE Prandtl number is 1; k is the turbulence kinetic energy and ε is the rate of dissipation. The pressure gradient, for a mean air velocity of 0.4 m s1, decreased by approximately 3 kPa m1, the equivalent of B 60 µm. Hence cell face distortion is the likely cause of the negative B values, if turbulent flow occurs. It converts a flat face (equivalent to an orifice in a flowmeter) into a

Hole Fixed at edge

Figure 8.7

Perspective view of 1/4 of a holed face, distorted by a pressure differential of 10 Pa on the upper face. The contour shades are of vertical displacement (Fitzgerald et al., 2004).

Table 8.3

Flow rates (106 m3 s1) for pressure gradient of 10 kPa m1 and cell diameter 1 mm Face shape Flat Distorted by 10 Pa pressure

Laminar flow

Turbulent flow

0.392 0.409

0.188 0.303


187

curved face (equivalent to a nozzle), with a consequent reduction in air-flow resistance.

8.3.3 Fourie and Du Plessis model Fourie and Du Plessis (2002) predicted air pressure drops through ‘Duocell’ open-cell aluminium foams, considering both streamline and turbulent air flow. They considered a representative unit cell (RUC) in which the edge length was the same as in a Kelvin foam of relative density R. The edges had uniform equilateral-triangle crosssections. The cumulative flow resistance of the edges, each treated as being in an infinite chamber, was considered; consequently they did not consider periodic boundary conditions. The predicted foam permeability K was related to the characteristic dimension d and the tortuosity T of the RUC by K

(1 R)2 d 2 36T (T 1)

(8.9)

For the Kelvin foam with uniform sized cells, d is approximately 57% of the cell diameter DC. Consequently, K depends on the square of DC and, for low R, is linearly dependent on R. A tortuosity of T 1.36 was taken from a model (Smit and Du Plessis, 1999) in which flow occurs down square cross-section pipes, which run successively in the x, y, and z directions. Acoustic methods of measuring tortuosity (Section 8.5) give similar values. The predicted pressure gradient vs. air velocity plots had a similar shape to those in Figure 8.3, but the pressure gradients were a factor of 8 smaller, due to the mean PU foam cell size being 20% of that of the metal foam.

8.3.4 CFD of Kelvin foam model CFD has been used to predict the permeability of a range of idealised foam microstructures. Fitzgerald et al. (2004) considered air flow in the [001] direction for a dry Kelvin model with circular holes in the faces, which can simulate the small holes in slow-recovery PU foams. Mills (2005) considered both [001] and [111] direction air flow in a wet Kelvin foam, an appropriate model for open-cell flexible PU foams. The air-flow permeability was found to be isotropic to within 1%, and the values confirmed those (for relatively large holes) of the circular hole model. Finally, Mills and Gilchrist (2006) considered air flow in the inverse of the wet Kelvin model, modelling air flows in channels in expanded polypropylene (EPP) foams. In all these models the air-flow

188 Polymer Foams Handbook Hole 1/8 square face

Hole

L/2

Symmetry plane

1/8 square face Hole Quarter square face

Flow

Half hexagonal face L

Hole

Periodic boundary condition Quarter on end faces Half square hexagonal face 1/8 face square face

Symmetry plane z

Figure 8.8.

Hole

Perspective view of Kelvin foam RUC, with front diagonal plane being transparent. The flow is periodic in the [001] lattice direction (Fitzgerald et al., 2004).

splits, merges, and changes direction, but there is no variation in cell shape and size. Due to the symmetry of the air flow in the [001] direction of the Kelvin model, it is only necessary to consider a triangular prism RUC (Fig. 8.8) of length 兹苶 2 L equal to the cell diameter between two parallel square faces. The two orthogonal prism sides have width 兹苶 2 L, while the diagonal side has width 2L, where L is the edge length. The prism contains 1/8th of a cell with a square face at the entry and exit of the unit, and two 1/16th cells, separated from the first by holes in half hexagonal faces, and from each other by a hole in 1/8th square face. The flow vectors have mirror symmetry at the prism sides and repeat periodically at the prism end faces. To mimic the real foam microstructure, the hole radii are the same fraction of the square and hexagonal face diameters (Fitzgerald et al., 2004). The pressure contour map (Fig. 8.9a) is the periodic part of the solution, after the constant pressure gradient has been removed. As there is glide plane symmetry between the left and right sides of the diagonal boundary plane, only the left side is shown in Figure 8.9a and the right side in Figure 8.9b. There is a nearly constant pressure differential of 4 Pa between cells on opposite sides of hexagonal faces, and 6 Pa across square faces perpendicular to the z-axis. The vector map (Fig. 8.9b) shows that the flow changes direction to pass almost perpendicularly through the hexagonal face holes. The flow velocity is highest at the face holes. The mean air velocity – V in the foam is obtained using equation (8.1) with A being the cross-sectional area of the prismatic unit and Q the input flow rate.

Chapter 8 Air flow in open-cell foams 3

2

1

189

0

0.4

1.2 0.8 3 −2 (a)

z 0

Figure 8.9

−1

−1

3

(b)

Laminar air flow under pressure gradient 10 kPa m1 on the diagonal side of Kelvin foam RUC with cut cell faces shown. Hole radius 0.2 L, cell diameter 0.6 mm: (a) contours (Pa) of periodic part of total pressure and (b) velocity vectors at every third mesh point and contours (m s1) (Fitzgerald et al., 2004).

The foam permeability K is computed from equation (8.2) without the V 2 term as K ηV

p L

(8.10)

where K was constant to within 2% for pressure gradients in the range 1–100 kPa m1, as expected for a laminar flow of a constant viscosity fluid. The predicted K values, for both the laminar and the turbulent flow, are lower than the Sunmate and Royal Medica data for the same hole area (Fig. 8.5). The data point for the Pudgee foam, with more uniform hole sizes (Fig. 2.8b), lies below the predictions. In the Kelvin model the holes in each cell are the same size. As K increases with hole area, a foam with a wide range of hole areas will contain some high permeability paths, hence have a higher K than a foam with a uniform hole size and the same mean hole area. The wet Kelvin model predicts, for a fixed foam relative density R 0.0622, that K has an S-shaped dependence on the cell diameter DC (Fig. 8.10). For the small cell diameters typical of cushioning


Permeability K (10−9 m2)

5

4

3

2

1

0

Figure 8.10

0

0.5

1 Cell diameter (mm)

1.5

2

Variation of K with cell diameter for wet Kelvin foams, with hexagonal face hole diameter 0.409 of the cell diameter, for a pressure gradient of 10 kPa m1 (Mills, 2005).

foams, K increases almost with the square of DC, as the velocity fields of the holes do not affect one another. For higher DC, where the velocity fields interact, there is a nearly linear region. This shows the limitations of models that ignore the periodic nature of the flow in foams. The predictions in Figure 8.10 are for smaller DC values than the experimental data in Figure 8.2.

8.3.5 CFD of bead foam channels In bead foams, air is trapped in the closed cells of the beads, and the volume fraction of free air in the inter-bead channels is low. The inverse of the wet Kelvin model, in which beads replace the air, and air replaces the solid PU, can be used for modelling. The bead diameter is typically an order of magnitude larger than the cell diameter in PU foams. Mills and Gilchrist (2006) considered air flow in the [001] lattice direction through the complex-shaped inter-bead channels (Fig. 8.11a). The predicted air velocity is maximum at mid-channel (Fig. 8.11b). To validate the CFD, air permeabilities were computed for an overlapping sphere model, in which the bead centres sit on a body centred cubic (BCC) lattice. Larson and Higdon (1988) used a boundary collocation method to analyse Stokes flow in such a model (Stokes flow is for very low Reynolds numbers, where the inertial


Bead

191

Periodic boundary

Mirror symmetry 0.8 Bead

0.6 0.4 0.2 z x

3 2

y 1

(a)

Figure 8.11

(b)

Perspective view of (a) RUC for wet Kelvin model with VA 0.085 and (b) CFD predicted air velocity contours with VA 0.062. The contours, at 20% intervals of the maximum velocity, are shown on the symmetry and periodic input plane of the air channel (Mills and Gilchrist, 2006).

effects in the differential flow equation can be neglected). They normalised their results by dividing the permeability by the square of the BCC lattice spacing d, since K scales with d2 for Stokes flow in periodic cubic models. Their results for the normalised permeability as a function of the model air volume fraction are compared in Figure 8.12 with the CFD result for the overlapping sphere model; their predictions, using a different analytical method, validate the CFD prediction for the overlapping sphere model. Their predictions fit the power law relationship K 7.51 103VA2.62 d2

(8.11)

with a correlation coefficient r 2 0.9999. The CFD predictions for the wet Kelvin model fall slightly below their data, as expected. Using equation (8.11), with mean bead diameter d 4.6 mm for the FPP3.0 foam and 3.1 mm for the FPP5.5 foam, and a porosity VA 0.25, the predicted permeabilities were 4.2 109 m2 and 1.9 109 m2, respectively. Thus the measured Brock foam permeabilities are, respectively, 18% and 27% of the predicted values.


K/d2 × 10−6

100

10

1

0.03

0.04

0.05

0.06 0.07 0.08 0.09 0.1

0.2

Volume fraction air channel (VA)

Figure 8.12

Normalised foam air permeability vs. volume fraction air channels: solid data points CFD results (Mills and Gilchrist, 2006).The trend line fits open data points from Larson and Higdon (1988).

8.3.6 CFD of the Weaire–Phelan model Boomsma et al. (2003) used the Weaire–Phelan minimum surface energy foam (Chapter 1) with triangular cross-section edges, a structure reasonably close to that of real open-cell foams. The foam geometry, generated using Surface Evolver, was exported to a CFD programme. They modelled a metal foam, with a mean cell diameter of 2.3 mm and relative density of 0.04. CFD, for slow (0.075 m s1) flow of water, predicted a pressure gradient that was 27% smaller than the experimental value for the foam. Their prediction corresponds to K 3.6 109 m2.

8.4

Air flow during foam impact compression 8.4.1 Air pressure changes during compressive impacts Jones and Fesman (1965) attributed a correlation, between air-flow and ball rebound values (for impact speed 3 m s1), to viscous air-flow losses in the foam, but did not model the phenomenon. Chapter 10 discusses a similar phenomenon in falls onto large mats of rebonded PU foam, where air flow may affect the response. Therefore, pressure variations were measured during impacts on such foams. Mills and

193

Chapter 8 Air flow in open-cell foams 35

20 Predicted

20

Measured

15 10 5

10

Measured

5 0 −5

0 −5 −5

Predicted

15

25

Pressure (kPa)

Pressure (kPa)

30

0

(a)

Figure 8.13

Table 8.4

5

10 15 20 Stress (kPa)

25

30

35

−10 −5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 (b) Stress (kPa)

Comparison of predicted and measured pressures vs. the mean foam stress, for 5 kg impacts at 4.4 m s1: (a) central stress in a 250 mm square block and (b) central air pressure in a 200 mm square block with a hole (Mills and Lyn, 2002). Slope of central pressure vs. foam stress, for loading impact tests Specimen width Central (mm) hole 100 150 200 250

No No No No

Drop height (m)

Experimental

Predicted

Experimental

Predicted

0.5 0.5 1.0 1.0

1.17 1.56 0.97 Oscillations

1.07 1.09 1.18 1.20

0.16 0.19 0.40 0.35

0.20 0.23 0.46 0.47

No hole

Central hole

Lyn (2002) used a flat-faced steel striker that fell vertically to impact rebonded PU foams, supported on a flat steel table; 100 mm tall foam blocks of length and width L, where L 100, 150, 200, or 250 mm, were prepared with and without a 13 mm diameter hole along their vertical central axis. The striker of mass 5.59 kg was fitted with a linear accelerometer. A 6 mm diameter quartz pressure transducer, fitted at the centre of the table, either measured the compressive stress on the foam lower surface or the air pressure in a central vertical cavity in the foam. Figure 8.13 shows the central surface pressure measured for a 1 m drop onto a 200 mm by 200 mm foam block, plotted against the average foam stress. There is a hysteresis loop for loading/unloading, with a nearly constant 14 kPa reduction in pressure on the unloading phase. Table 8.4 gives values for the other specimen sizes.

8.4.2 Air-flow modelling The two extreme assumptions are either that air flow is unrestricted by the foam, so it has no effect on the compressive stress, or that air

194 Polymer Foams Handbook is trapped in the foam cells, so pneumatic compression is the main contribution to the stress in the relatively soft foam. Modelling will show whether or not it is acceptable to make either assumption, or whether an intermediate state exists. There is a parallel with the analysis of air compression (isothermal or adiabatic) in compressed closed-cell foams, discussed in Chapter 11. 8.4.2.1

Axisymmetric model

Mills and Lyn (2002) modelled a flat-based striker of mass m and initial velocity V0, hitting a foam block and compressing it uniaxially. The mass has a single degree of freedom, the y co-ordinate of its lower flat face. The axisymmetric model approximates the geometry of experimental samples, which were blocks of width and length L (Fig. 8.14). The vertical foam cylinder, of diameter D L, is split into 20 concentric annuli of thickness ∆r. The foam undergoes uniaxial compression between two flat steel anvils. It is assumed that: (a) The foam strain is homogeneous. Sims and Bennett (1998) observed bulging during the impact of PU foam specimens with large central voids, but not deformation bands. If the strain is uniform in the y direction, there is no air flow in that direction. (b) The polymer is not rate dependent (viscoelastic). (c) The striker surface, on contact, instantly seals the foam upper (and lower) surface. This causes the air flow to be purely in the

Mass m

V y

h Pi −1

D (a)

Figure 8.14

(b)

Pi

∆Vi

Pi +1

∆Vi +1

Finite difference axisymmetric models of foam cylinder impacted at velocity V by a flat striker, while resting on a flat table: (a) model geometry and (b) three annuli contain air at constant pressure, while the connecting pipes provide flow resistance (Mills and Lyn, 2002).


195

radial r direction. Cut edges readily fold over, so the air-flow resistance of partial cells at the surface should be higher than other parts of the foam. (d) The foam Poisson’s ratio is zero, a good approximation for compressive strains exceeding 10%. (e) The conditions are isothermal, due to forced convection air flow past the cell edges. Even for closed-cell foams with a sub-mm cell size, compressed at 5 m s1, Chapter 11 shows the conditions are nearly isothermal. (f) The initial impact velocity is small compared with the speed of sound through the foam, so sound pressure waves can be ignored. Voronina (1998) gave the sound velocity in PU foam of density 32 kg m3 as 190 m s1, for frequencies of 0.5–2 kHz. The time for such a wave to rebound through a 0.1 m thick foam is about 1 ms. There are signs, on the post-collapse impact stress–strain graphs, of damped pressure oscillations on a slightly shorter scale. 8.4.2.2

Explicit finite difference calculations

The model, using the air-flow equation (8.2), assumes that the flow resistance only depends on the pressure differential, and not on the absolute pressure. Hence the model is not applicable when the pressure is much greater than atmospheric. The annuli decrease in volume as the foam is compressed in the y direction. The finite difference approximation considers annulus i as a reservoir of gas at a constant pressure pi, and localises the gas flow resistance to intervening ‘pipes’ (Fig. 8.14b). The time interval ∆t between steps was the longest that was stable; typically 1 µs for a 200 mm diameter cylinder and 0.1 µs for a 100 mm diameter cylinder. The main calculation steps, repeated at intervals ∆t, are the following: 1. Strain increase, due to the striker velocity acting for the time interval ∆t. 2. Foam air-flow parameters for the new strain, using equations (8.5) and (8.6). 3. Pressure increase in each annulus, assuming no change in the gas content, using the ideal gas law pV/T constant

(8.12)

where p is the absolute pressure, V the volume, and T the absolute temperature. 4. Flow velocity vi through the pipe to the right of the annulus i, calculated from the pressure gradient ∆pi pi 1 pi using an inversion of equation (8.2). The sign of v is the same as the sign of ∆pi.

196 Polymer Foams Handbook 5. Net volume gain ∆Vi of each annulus, in the time interval, due to the flow velocities ∆Vi ri1 vi1 ri vi

(8.13)

6. New annulus pressures pN, from the net volume gain, using ⎡ V t ⎤⎥ pN (i) pO (i) ⎢⎢ 1 r ri ⎥⎥⎦ ⎢⎣

(8.14)

7. The annulus pressures were stored as ‘old’ values pO for the next time increment. 8. Mean air pressure, across the foam upper surface, hence, adding the contribution from the PU structure, the mean compressive stress in the foam. 9. Impact force as the product of the foam mean compressive stress and its top surface area, hence the striker deceleration, using Newton’s second law. 10. New striker velocity, from the deceleration and time interval. 11. Height h of the foam block, from the striker velocity and the old height. 12. Energy input (output) as a numerical integral of the striker force vs. the foam deflection when the deflection was increasing (decreasing). 8.4.2.3

Predictions

Figure 8.15 shows the pressure distribution, across a 500 mm diameter foam cylinder, is almost an inverted parabola during loading. The relative pressure at the cylindrical surface is always zero. However, during unloading when the mean pressure is low, more complex S-shaped distributions can occur. Later, there is a positive parabolic distribution during the rebound phase. Figure 8.16 shows predicted stress–strain curves for D 100 and 250 mm, for 64 kg m3 density PU chip foam. The comparison curves are for: (a) No gas loss from a foam of zero Poisson’s ratio, and zero polymer contribution, so the applied stress is given by σ

paε 1 ε R

(8.15)

where pa is atmospheric pressure. This acts as an upper limit.


197

60 6 4

40 Stress (kPa)

8 2

20

10 0

14 12

−20.

0

62.5

125

187.5

250

Distance from midlayer (mm)

Figure 8.15

Predicted pressure profiles, at the labelled times (ms), across a 500 mm diameter foam cylinder, impacted by a 5 kg mass at 10 m s1 (Mills and Lyn, 2002).

40

250 No gas loss

30

Ogden

Stress (kPa)

Stress (kPa)

200 No gas loss 20

150 100 50 Ogden

10

0 −50

0 0

10

(a)

Figure 8.16

20

30

40


50

60 (b)

0

20

40

60

80


Predicted compressive stress–strain curve for 100 mm high cylinders using K0 5.4 109 m2 and B0 130 µm: (a) 250 mm diameter cylinder at V 4.4 m s1 and (b) 500 mm diameter at V 10 m s1. The comparison curves are for a ‘ideally soft’ foam with no gas loss and Ogden strain energy function (Mills and Lyn, 2002).

(b) The uniaxial compressive stress–strain curve, calculated from the Ogden strain energy function using the parameters µ1 18 kPa, µ2 1.2 kPa, α1 8, α2 2, ν1 0, and ν2 0.45. This acts as the lower limit for a model with no heat transfer. As the foam block diameter and impact velocity increase, the predicted mean stress increases relative to the Ogden function lower limit. For

198 Polymer Foams Handbook a 100 mm diameter cylinder impacted at 3.1 m s1, the predicted airflow contribution is negligible. For the 250 mm diameter foam cylinder impacted at 4.4 m s1 (Fig. 8.16a), the air flow increases the stress during loading by about 50%. For a 500 mm diameter foam cylinder impacted at 10 m s1, the predicted loading curve (Fig. 8.16b) approaches the ‘closed-cell’ upper limit. The model predicts that the hysteresis magnitude, in graphs of the central foam stress vs. the mean foam stress (Fig. 8.13a), is nearly independent of the specimen size. This is confirmed by experimental data. For foam cylinders with a small central hole, the predicted slope of a graph of central (air) pressure vs. the mean foam stress, during loading, increases from 0.20 to 0.47 as the cylinder diameter D increases (Table 8.4). The values are slightly larger than the experimental slopes. For foam blocks with no central hole, the predicted slopes are within 0.3 of the experimental values. Experimental loading slopes 1 are consistent with the pressure distribution across the surface of the foam being an inverted parabola, of the type shown in Figure 8.15. Figure 8.13a compares predictions with experimental data for the 250 mm square sample. The predicted hysteresis, expressed as the pressure differential between loading and unloading, half-way to the maximum stress, is 60% of the observed values. Consequently, most of the experimental hysteresis must be due to viscoelasticity, rather than gas flow resistance. Foam stresses are only high when the compressive strains are very high, so significant air-flow effects are high strain phenomena. The high strain (ε 0.7) values of B and K are critical to the success of the predictions.

8.5

Sound absorption in foams Some foams are used as sound absorption materials. If smooth, paintable surfaces are not required, egg-crate surfaced PU foams are effective sound absorbers; they are used in anechoic chambers. The empirical design of sound absorbers is described by Hamdi et al. (2001). In this section the meaning of the acoustic design parameters is explored. Acoustic engineers often work with the properties of a structure, rather than material properties. The relationship, between acoustic properties at an angular frequency ω and steady state air-flow measurements, is similar to the relationship between the dynamic mechanical response at a frequency ω and a tensile test. The foam permeability K is equivalent to an acoustic property at ω 0, but further frequency-dependent properties are needed. As sound waves cause small air displacements,


199

the response should be linear, without a second-order term B (inertial flow coefficient). The factors that could affect the sound absorption in open-cell foams are as follows: (a) Porosity, defined as 1 R. (b) Permeability K. (c) Viscous characteristic length Λ that relates to the high frequency response. (d) Polymer modulus and internal damping (tan δ). However, all flexible PU foams may have sufficiently low Young’s modulus and sufficiently high tan δ to be optimal. (e) Shape of the foam block related to the supporting panel or tube. There is no equivalent of CFD to compute acoustic parameters from a foam model. Bolton and Kang (1997) state that materials, that have a much higher bulk modulus than air, can only propagate a single longitudinal wave. This condition appears to be satisfied for air in polymer foams. The tortuosity (given the symbol α in acoustics papers) causes the wave velocity Cf in the foam to be smaller than that (C) in air α∞

C Cf

(8.16)

The tortuosity is defined as the ratio of the mean square molecular velocity in free air, to that in the air inside the foam. The subscript

indicates extrapolation to infinite frequency. This definition differs from that given in Section 8.3.3. Melon et al. (1998) measured the tortuosity of a PU foam, of unspecified cell size, using the dispersion of the wave speed, as 1.24 parallel to the rise direction and 1.32 perpendicular to the rise direction. However Ayrault et al. (1999) measured values of only 1.05 for Recticel foams of large cell size. As some acoustic parameters cannot be related to the foam microstructure, microstructural targets cannot easily be set for improved acoustic foams. Voronina (1998) proposed an empirical model in which sound propagation is affected by viscous losses in the air, which are functions of the pore diameter and the porosity, and the losses in the polymer, which depend on the compressive stress to achieve 20% strain. Cummings and Beadle (1993) show, for ducts with PU foam liners, that the axial attenuation rate (Fig. 8.17) is greater for foams with large K at low frequencies, but greater for foam with small K at high frequencies. The change-over occurs in the range 30–200 Hz. Since K increases with the square of the cell diameter, large cell foams are preferable at low frequencies.

200 Polymer Foams Handbook 100 10,000 5000 2500 1250 625

10 ∆ (dB/m)

312

50 mm

1

100 mm

Duct

50 mm

PU foam absorbent 0.1 10

Figure 8.17

100 Frequency (Hz)

1k

Sound attenuation as a function of frequency for foam flow resistivity values (rayl m1) shown (Cummings and Beadle, 1993 © Academic Press).

Closed-cell polyethylene (PE) foams have advantages for sound absorption in environments (e.g. boats) where water ingress is unwanted. PU foams suffer from hydrolysis and can act as sponges. Dow ‘Blucell’ foam is a large cell sized PE foam. Park et al. (2001) describe the development of acoustical polyolefin foams, showing that partial perforation of the cells improves the sound absorption at low frequencies. Dow ‘Quash’ closed-cell PE copolymer foams are shown (www.acoustop.com) to have a better resistance to water exposure for 5 weeks than PU open-cell foam with a protective film coating. Closed-cell foams, with a large cell size, based on blends of PE and polypropylene, can have a maximum attenuation at a frequency circa 500 Hz, due to a membrane absorber of the cell faces.

8.6

Filters Since the permeability of open-cell PU foams with coarse cells is high, these have been used as air filters in vehicles, and in anaesthesia, where


201

both water and air transfer are required. They have also been coated with ceramic for use as filters in aluminium casting, and coated with metals for use in heat transfer units. Turnbull et al. (2005) compare the performance of PU foams with other filter media for anaesthesia. There is some increase in the pressure drop across PU foam filters under wet conditions. Foams with smaller mean cell diameter take up less moisture. Kavouras and Koutrakis (2001) assessed PU foam as particle size analysers, when used at high Reynolds numbers (2000–7000). They were able to collect dust particles with diameters as low as 0.4 µm. There is increasing interest in using PU foam as a medium for absorbing biological contaminants. Moe and Irvine (2000) describe the use of PU foam as a gas phase biofilter. If the PU foam is used in a water flow, it can support silver nanoparticles, and act as an antibacterial filter (Jain and Pradeep, 2005).

References Ayrault C., Moussatov A. et al. (1999) Ultrasonic characterization of plastic foams via measurements with static pressure variations, Appl. Phys. Lett. 74, 3224–3226. Bhattacharya A., Calmidi V.V. & Mahajan R.L. (2002) Thermophysical properties of high porosity metal foams, Int. J. Heat Mass Trans. 45, 1017. Bolton J.S. & Kang Y.J. (1997) Elastic porous materials for sound absorption and control, SAE Trans. J. Passen Car. 106, 2576. Boomsma K., Poulikakos D. & Ventikos Y. (2003) Simulations of water flow through open-cell metal foams using an idealised periodic cell structure, Int. J. Heat Fluid Flow 24, 825–834. BS 4443-6 (1991) Methods of Test for Flexible Cellular Materials: 16 Determination of Air-Flow Value, British Standards Institution, London. Cummings A. & Beadle S.P. (1993) Acoustic properties of reticulated plastic foams, J. Sound Vib. 175, 115–133. Fitzgerald C., Lyn I. & Mills N.J. (2004) Air flow through polyurethane foams with near-circular cell face holes. J. Cell. Plast. 40, 89–110. Fluent Inc (2003) Modelling turbulence, Chapter 10, Fluent Manual, Fluent Inc, Lebanon, USA. Fourie J.G. & Du Plessis J.P. (2002) Pressure drop modelling in cellular metallic foams, Chem. Eng. Sci. 57, 2781–2789. Gent A.N. & Rusch K.C. (1966) Permeability of open-celled foamed materials, J. Cell. Plast. 2, 46–51.

202 Polymer Foams Handbook Hamdi M.A., Mebarek L. et al. (2001) An efficient finite element formulation for the analysis of acoustic and elastic wave propagation in sound packages, SAE Noise and Vibration Conference, Michigan, Paper 165. Hilyard N.C. & Collier P. (1987) A structural model for air-flow in flexible PU foams, Cell. Polym. 6, 9–26. Jain P. & Pradeep T. (2005) Potential of silver nano-particle coated polyurethane foam as an antibacterial water filter, Biotech. Bioeng. 90, 59–63. Jones R.E. & Fesman G. (1965) Air-flow measurement and its relations to cell structure, physical properties and processibility for flexible polyurethane foam, J. Cell. Plast. 1, 200–216. Kavouras I.G. & Koutrakis P. (2001) Use of polyurethane foam as the impaction substrate in conventional inertial impactors, Aerosol Sci. Tech. 34, 46–56. Kozeny J. (1927) Uber kappillare Leitung des Wassers im Bodem, Akad. Wiss. Wien Ber. 136, 271–306. Larson R.E. & Higdon J.J.L. (1988) A periodic grain consolidation model of porous media, Phys. Fluid. A1, 38–46. Melon M., Mariez E. et al. (1998) Acoustical and mechanical characterization of anisotropic open-cell foams, J. Acoust. Soc. Am. 104, 2622–2627. Mills N.J. (2005) Modelling of airflow through open-cell polyurethane foams, J. Mater. Sci. 40, 5845–5851. Mills N.J. & Gilchrist A. (2006) Properties of bonded-polyproplenebead foams: data and modelling, J. Mater. Sci. (accepted). Mills N.J. & Lyn G. (2002) Modelling of air-flow in impacted flexible polyurethane foams, Cell. Polym. 21, 343–367. Moe W.M. & Irvine R.L. (2000) Polyurethane foam medium for biofiltration, J. Environ. Eng. 126, 815–825. Paek J.W., Kang B.H. et al. (2000) Effective thermal conductivity and permeability of aluminium foam materials, Int. J. Thermophys. 21, 453–464. Park C.P., Burgun S. et al. (2001) Novel acoustic polyolefin foams, SAE Noise and Vibration Conference, Michigan. Ruth D. & Ma H. (1992) On the derivation of the Forchheimer equation by means of averaging theorem, Transp. Porous Media 7, 255–264. Scheidegger A.E. (1960) The Physics of Flow Through Porous Media, University of Toronto Press, Toronto, Canada. Sims G.L.A. & Bennett J.A. (1998) Cushioning performance of flexible polyurethane foams, Polym. Eng. Sci. 38, 134–142. Smit G.J.F. & Du Plessis J.P. (1999) Modelling of non-Newtonian purely viscous flow through isotropic high porosity synthetic foams, Chem. Eng. Sci. 54, 645.


203

Turnbull D., Fischer P.C. et al. (2005) Performance of breathing filters under wet conditions, Br. J. Anaesth. 94, 675–682. Voronina N. (1998) An empirical model for elastic porous materials, Appl. Acoust. 55, 67–83. Zhang X.D., Macosko C.W. & Davis H.T. (1997) Effect of silicone surfactant on air flow of flexible polyurethane foams, in Polymeric Foams, Ed. Khemani, ACS Series 669, p. 130.

Chapter 9

Seating case study

Chapter contents 9.1 Introduction 9.2 Biomechanics of sitting in chairs 9.3 Car seats 9.4 Foam selection 9.5 Seat design 9.6 Other foam mechanical properties Summary References

206 206 214 217 225 228 230 230


9.1

Introduction This chapter explores the biomechanics of sitting, and the foams available to make seats comfortable. The emphasis is on car and wheelchair seating, on which more has been published than on domestic or office seating. The design of wheelchair seating is particularly important, as the users remain seated for long periods, and may not be able to adjust their position. The able-bodied, if they become uncomfortable, can shift their seating position. The biomechanics of pressure sores, and the interaction between human tissue and foam seating, are introduced first. Body posture is considered, since it affects the pressure distribution on the surface of the buttocks. Pressure mapping technology can provide real-time indication of pressure ‘hot-spots’ and allows the correction of posture, and the selection of optimal foam seating. Criteria for comfort are complex, involving factors such as moisture transmission and heat flow. The second part of the chapter considers the mechanics of foam seating. The requirements on vehicle seating are more complex than those for domestic seating, since vehicle vibrations are partly transmitted via the seating to the occupants, and the seat plays part of the crash protection system. Complex-shape, moulded car seats require different polyurethane (PU) formulations and processing equipment to achieve low cycle times; consequently the foam structure and properties differ from slabstock PU foam used in domestic chairs and bedding.

9.2

Biomechanics of sitting in chairs 9.2.1 Seating posture and mannikins Lueder (1994) considered the ergonomic design of office chairs, often used for computer operation. The lower spine should be S rather than C shaped for comfort, which means that the pelvis should be rotated forwards. The ideal position in an office chair is quite different from the semi-recumbent position in car seats. The latter uses a seat rear angle of 20° or more from the vertical, so that the roof-level can be low (Fig. 9.1). The height of the seat above the car floor is usually smaller than that of a domestic chair seat (typically 0.45 m for a dining chair and 0.4 m for a sofa). The cervical spine is of necessity bent forwards, and the driving position is maintained without change for long periods; the chance of the spine becoming stiff is greater than with an upright chair. The position of the hip (H point) affects whether the driver has an adequate field of vision, so it must be at a specified distance below the

Chapter 9 Seating case study

Figure 9.1

207

Semi-recumbent seating position adopted in most cars, illustrated by the ‘John’ dummy (Hubbard et al., 1993 © SAE).

windscreen. Hubbard et al. (1993) described anthropometric models that represent the extremes of the car driving population. However, dummies sit in cars in unrealistic postures (Stabler et al., 1996); drivers tend to sit further forwards, to better observe the outside world. Anthropometric databases, such as Peoplesize, contain the dimensions of different populations, but insufficient detail for car seat design. Figure 9.2 shows the pelvis anatomy.

9.2.2 Pressure sores and ischaemia When sitting on a flat wooden bench, the peak pressures on the buttocks are higher than when sitting on a soft foam cushion. Such high pressures are tolerable for short periods, and can be relieved by changing position. For those confined to bed or a wheelchair, there is a risk of pressure sores (pressure ulcers). Ischaemia means the blood supply, from the capillaries of the blood circulation system to the muscles, is cut off when blood vessels occlude (close). Surface pressures can close the capillaries in the muscles, in a similar way to an inflated cuff on the upper arm cutting off the venous blood flow, when blood pressure is measured. When the blood circulation stops, toxic breakdown products of cell metabolism build up in the tissue, and may eventually cause cell damage (Brienza and Geyer, 2001).


(a)

Figure 9.2

(b)

(a) Side and (b) frontal view of the pelvis of a seated person, with ischial tuberosities arrowed.

Research prior to 2000 tended to emphasise pressure as the cause of pressure sores. The technology for measuring pressures on the skin was readily available (see the next section). Ferguson-Pell (1990) related the risk of pressure sores to the peak skin pressure values under the seated patient, shear stress, patient factors (age, nutrition, blood pressure, smoking, and mobility), and environmental factors (moisture). The term pressure in ‘pressure sore’ may confuse engineers, because hydrostatic pressure does not cause ischaemia. Soft tissues are mainly water, which has a high bulk modulus of about 2 GPa; 100 kPa hydrostatic pressure, at a depth of 10 m in the sea, neither causes soft tissue shape change nor affects the blood supply. The surface pressure is a uniaxial compressive stress, acting perpendicular to the skin surface. If this pressure acts on part of the skin, the soft tissue changes shape. Ischaemic events can also occur some distance below the skin, near bony protuberances; the stress state at such sites is more complex than uniaxial compressive stress. The relevant bony protuberances for sitting are the ischial tuberosities (Fig. 9.2). Aissaoui et al. (2001) measured their length as up to 8 cm with width 2–4 cm. The sacrum, at the base of the spine, can also cause a pressure peak. Goosens et al. (1994) measured the skin oxygen tension with a small electrode, when compressive and shear stresses were applied to the backs of healthy young volunteers. Their systolic and diastolic blood pressures, respectively, averaged 15.9 and 10.7 kPa (the pressures to stop and to re-establish the blood flow). They defined the cutoff surface pressure as that which reduced the skin oxygen pressure to 1.3 kPa, from a starting value of around 10 kPa. The cut-off pressure of 11.6 kPa, with no shear stress, reduced to 8.7 kPa when the shear stress was 3.1 kPa. Hence shear stress causes ischaemia, as well


209

as compressive stress. Both the vein wall compliance, and its rate of shape recovery after occlusion, reduce with age, as the proportion of collagen to elastin in the vein wall increases. Therefore the tolerance to surface pressure reduces with increasing age. Bouten et al. (2003) criticised the traditional quoted pressure threshold of 4.3 kPa for tissue damage. They stated that capillary closure depends on local pressure gradients across the vessel wall, and not just on pressures on the skin. They argued that deep pressure ulcers, which can occur at the bone/muscle interface, may be caused by tissue deformation rather than pressure (Section 9.5.4 returns to this topic). Magnetic resonance imaging was used (Stekelenburg et al. 2006) to detect tissue damage in rats due to local indentation of muscle; this was intended to simulate the indentation of muscle tissue by bony prominences.

9.2.3 Measuring seating pressure distributions In spite of the debate on the causes of pressure sores, a nearly uniform skin pressure distribution is an aim of seating design. Seating pressure distributions depend on the cushion shape and material, and the individual’s weight, pelvis shape, and soft tissue thickness. The pressure measurement system should fit between the cushion and the buttocks, without influencing the pressure distribution. The ideal system should be: (a) Thin, the rubber sensors in FSA mats are 2 mm thick, while those based on conductive ink between PET films can be 0.5 mm thick (Tekscan, 2000). (b) Flexible, so that the bending resistance does not cause load spreading. Both of the above systems meet this requirement. (c) Extensible in the plane, with similar coefficient of friction as clothing on a cushion. With friction between the mat and the foam, the sensor mat’s resistance to extension significantly increases the cushion’s indentation resistance, and changes the pressure distribution. The effect should be smaller with smooth film systems. (d) Present with a low thermal barrier, to allow the cushion to reach typical temperatures. Modern systems, with a high number of sensors, allow real-time display of pressure maps. Podoloff (1993) described a pressure-sensing mat, using the resistance change in a carbon-filled polymer layer. An array of 2000 sensors on a 1 cm grid spacing detected pressures up to 20 kPa at frequencies up to 100 Hz. None of the sensors are ideal, since their response is slightly time dependent. Figure 9.3a shows pressure contours using the FSA system on a thin foam cushion (100 mmHg ⫽ 13.3 kPa). The aim is a uniform pressure with values


Sensors included Variation coefficient Standard deviation Average pressure Maximum pressure Centre of pressure

(a)

Figure 9.3

166 92.9% 26.5 26.5 158 8.3, 7.5

(b)

(a) Pressure contours for a student on 25 mm of PU foam, with the knees at the top of the figure, and the peak pressures under the ischial protuberances and (b) 3-D pressure map (Mills, unpublished).

less than 12 kPa. Pressure distributions are also relevant to the comfort of the general population sitting or lying on foam products.

9.2.4 Comparative deformation of the thigh and foam cushion The deformation of the leg or buttock depends on the pressure distribution on the skin and on the muscle activity. Setyabudhy et al. (1997) measured the shape of the thigh/seat interface when volunteers sat on a 41 kg m⫺3 density PU foam cushion, with an initially flat upper surface. The subject’s knee and hip were fixed, so their femur did not move. The results depended on the knee angle; the lower the value, the higher the tension on the hamstring muscles on the rear of the thigh. The viscoelasticity of foam and the thigh was ignored, with measurements made 30 s after raising the seat. Vertical wires through the 100 mm thick cushion indicated the position of the leg surface. The thigh deformation, found by subtracting the foam deformation from the total deformation, was larger than the foam deformation (Fig. 9.4). However, for forces exceeding 60 N, the foam stiffness exceeded the thigh stiffness. Both materials are non-linear, but the Poisson’s ratio of foam is much lower than that of the thigh. The initial skin to femur distance can be estimated from data in the Peoplesize database as about 70 mm. Hence, the maximum compressive strain in the thigh tissue, when the deflection is 40 mm, is about 57%.

Chapter 9 Seating case study 140 Foam

Thigh

211

Total

120

Force (N)

100 80 60 40 20 0

0

10

20

30

40

50

60

70

80

Vertical deformation (mm)

Figure 9.4

Deformation of the thigh and seating cushion vs. applied force. Knee angle: 100°, etc.; foam of density: 41 kg m⫺3 (redrawn from Setyabudky et al., 1997 © SAE).

Brienza et al. (1996) showed that the pressure vs. indentation relation of the buttock varied considerably with position, being stiffest when the pressure was applied, via a 120 mm by 120 mm plate, below the ischial tuberosities. However the soft tissue thickness was not measured. If the soft tissue dimensions, surface pressure, and skin deflections were all monitored, subsequent finite element analysis (FEA) would allow better material parameters to be derived.

9.2.5 Moisture and heat transmission to the seat Most car seats have a laminated cloth-faced layer on top of the main PU cushion (Fung and Parsons, 1996). A 3–6 mm thick PU foam layer is flame laminated to a polyester cloth face, and to a nylon scrim backing. This makes the cloth less likely to wrinkle, and allows moisture to pass through the cover. However some covers can act as a barrier to perspiration. A wool-cloth face to the laminate could be hygroscopic, absorbing water vapour, so making the laminate uncomfortable. Polyester fabrics were treated to be hydrophilic; this increased the perceived comfort. The car seat is in contact with about 25% of the body surface area, so acts like an extra layer of clothing, providing thermal insulation. A thick layer of foam in the laminate increases insulation, hence the

212 Polymer Foams Handbook skin temperature. Ferguson-Pell (1990) showed that water-filled wheelchair seats could transfer heat away from the body, while a layer of gel on top of the seat cooled the skin at first, but had a neutral effect after 2 h. A foam seat could cause the skin temperature to rise by 9° after 2 h.

9.2.6 Design of wheelchair seats A range of materials is used in seat cushions (Brienza and Geyer, 2001): foams, polymer gels, inflatable air pockets, soft rubber honeycombs, and combinations of these materials. Conventional seat cushions contain layers of slabstock PU foams, whereas specialised ones contain fluid or air chambers in which the pressure can be adjusted. Foam cushions can be contoured to suit the patient. The canvas seat of many wheelchairs, attached to the metal frame, also plays a part in the seating mechanics; it provides a better skin pressure distribution than does a rigid flat surface. In some specialist chairs, the support frame can be moulded, or adjusted via a large number of links, to achieve the required postural position of the occupant. For a particular foam type, the cushion thickness and compressive modulus can be chosen, if the weight of the patient is known (Fig. 9.5). The intention is to achieve an average compressive strain of about 50%. The cushion thickness should exceed 37 mm, so that there is sufficient contact area with the buttocks, but be less than 100 mm; a X-firm

Firm

Medium

Soft

1200 Bottoms out

Body weight (N)

1000 800 600

Unstable position

400 200 0

0

20

40

60

80

100

120

140

Cushion thickness (mm)

Figure 9.5

The cushion thickness is recommended to be in the pale area (redrawn from Sunmate brochure).


213

thicker cushion would be bulky, and might allow undesirable postures to develop. A range of cushion types can be tried, with real-time pressure mapping, to see which gives the best pressure distribution. It is possible to adjust the shape of a foam cushion, by cutting away areas, to minimise the pressure maxima under a patient. Several iterations may be needed before the cushion shape is optimum. Brienza et al. (1996) found that, when a wheelchair cushion shape was optimised for the occupant, the pressure maximum over the seated area could be reduced to 12 kPa. However, it is unrealistic to expect a uniform skin pressure distribution under a seated person; the anatomy of the buttocks will always cause non-uniform pressure distribution. Some possible cushion shapes for wheelchair users are shown on the foamstudies.bham.ac.uk website in the Seat Cushions/experiments section. It is possible to sculpt the upper surface contours of the foam to improve the pressure distribution. The foam can be segmented; vertical slots, in the upper part of the foam, can divide it into strips (as in the outer layers of some German mattresses) or into blocks (as in some wheelchair cushions). The shear stress σxy is zero on the plane of these slots, so long as the friction coefficient is zero – wider slots can reduce the friction, but may reduce the stability of the foam pillar. At the slots the horizontal tensile stress is zero. In a non-segmented cushion, these stresses contribute to the indentation H value (Section 6.2.4), so the modified seat should have a lower value of H. By reducing the vertical shear stress σxy to zero, the complimentary shear stress σyx is also zero at the slot surfaces. Since the shear stress σyx on the skin surface contributes to the risk of ischaemia, this is another advantage of cushion segmentation. The continuous lower part of the cushion holds it together.

9.2.7 Mattresses and sleep comfort Hospital operating tables consist of a thin mattress over a flat rigid table. If the mattress is unsuitable, there is a risk that the patient, anaesthetised for a few hours, will develop pressure sores. Vakeva (1997) considered the relation between pressure maxima and the foam construction of domestic mattresses. For a person lying on their side, the shoulder and hip cause pressure peaks. He found that multi-layer foam mattresses gave the ‘best’ results. Bader and Engdal (2000) showed that personal preferences, for a hard or a soft mattress, affected the ranking of comfort in beds. Their research monitored the number of times the sleeper turned in the night, as well as classifying four stages of sleep using electroencephalography and electromyography. Even with experiments lasting seven nights, it is not clear that the subjects had adapted to the new mattress types. Hence, decisions to

214 Polymer Foams Handbook change to novel forms of foam mattress, for instance by using lowresilience overlay, should not be based on a short trial.

9.3

Car seats 9.3.1 Types of car seat According to Gurram and Vertiz (1997), the general standard of automotive seating has improved to the point that local hard points no longer occur. Traditional seat bases have a steel frame, with horizontal steel coil-springs at the perimeter, supporting a flexible grid of steel wires, covered with a 25 mm thick foam cushion. There is a move towards all-foam seats; half of Japanese automobiles, and a growing proportion of American cars, have deep foam seating (Polyurethane Foam Association (PFA), In Touch, 1997). Such seating could have reduced mass and cost. ICI Polyurethanes (now Huntsman) in Europe aimed to develop deep-foam seats of thickness 70 mm, compared with 130 mm thick first generation full-foam seats, while maintaining seat durability and occupant comfort (Tan, 1996). Seat design depends on customer preferences; in Germany harder seats are used, with smaller static compression strains than in the UK (Pywell, 1996). In some luxury car seats (Mercedes) the whole cushion rides on a mechanical spring. In cheaper lighter seats, the foam cushion provides the majority of the deflection; it changes shape significantly under load. The force–deflection relationship of the Mercedes seat is more nearly linear than that of a foam block seat. Blair et al. (1999) correlated 16 mechanical properties of car seat foams, from all parts of the world. The Japanese market uses foams with high energy dissipation and moderate strength, the North American market uses foams with low energy dissipation, while the European market is intermediate. Density variations are possible in a single seat moulding. This allows higher density (stiffer) foams at the side wings of seats. When the car corners, the lateral acceleration can be 0.8g, so lateral support is necessary to prevent the person sliding sideways on the seat.

9.3.2 Comfort Milivojevich et al. (2000) studied two best-in-class and three luxury vehicle seats, using subjects who varied in height, weight, and sex. There was a correlation between high psychometric comfort scores while driving and low values of the mean peak ischial pressure, low mean thigh pressure, and high cushion contact area (Table 9.1). Andreoni et al. (2002) correlated seating pressure distributions with posture; some drivers achieved a uniform pressure across the trochanters and ischial tuberosities, while others had ischial pressure peaks.


Table 9.1

215

Comfort scores and body pressure distributions

Seat 1 2 3 4 5

Comfort score

Mean ischial pressure (mmHg)

Mean thigh pressure (mmHg)

Total contact area (cm2)

77 67 69 57 53

34 27 30 26 25

1124 1220 1166 1357 1316

1.0 4.5 2.1 6.7 10

6 A

Person of mass (m )

5

A/A0

4

n

Model 3

k 2 Data 1

A0

Vibrating car body

(a)

Figure 9.6

0

2

(b)

4

6

8

10

Frequency (Hz)

(a) Model of a mass m on a damped spring of constant k and (b) transmissibility of a car seat against frequency, data compared with model response (redrawn from Kinkelaar et al., 1998 © Sage).

9.3.3 Vibration transmission The seat provides isolation from both lateral and vertical acceleration of the car body. Figure 9.6a shows the transmissibility, defined as the seat surface vertical amplitude divided by the shaker amplitude, of a (European) car seat against frequency. There is a resonance at 4 Hz and partial vibration isolation at higher frequencies. The resonance occurs at 2.5–3 Hz in US cars (Cavender and Kinklear, 1996). If the seat transmits large amplitude vibrations in the frequency range of 2–20 Hz, the passengers may be sick – the stomach has a resonant frequency in this range. Griffin (1990) linked motion sickness in cars to horizontal vibration of the head, affecting the ear and balance

216 Polymer Foams Handbook system, rather than to the amplitude of vertical head oscillations. He found that many car seats transmitted similar levels of body acceleration in the frequency range 0.1–1 Hz. Therefore, the avoidance of nausea in young passengers may lie with them looking outside, so their brain can estimate their head motion, and the driver minimising lateral acceleration. The simplest model for the vibration isolation of a car seat uses a body mass m on a seat spring of constant k, with a parallel damper of constant n. The system has a single degree of freedom, the vertical position of the mass. If the damping constant is relatively low, the system resonant frequency ω0 is given by ω0 ⫽ 2π f0 ⫽

k m

(9.1)

The response to a driving frequency ω is shown in Figure 9.6b. Highdamping foam reduces the transmissibility, defined as A/A0, at and below the resonant frequency. All commercial PU foams have significant damping, so the seats do not cause high-amplitude resonance. If the driving frequency exceeds the resonant frequency, there is less transmission when the foam damping is low. The isolation factor improves markedly with increased frequency, if the frequency is above the resonant value. Soft seats are used to reduce the resonance frequency. The transmissibility of a car seat is a function of the amplitude of the applied acceleration, whereas the model shown in Figure 9.6 is amplitude independent. Patten et al. (1998) used a non-linear stress–strain model for PU foam, and managed to predict the amplitude dependence of the transmissibility. There may be multiple cycles of similar magnitude, when a car passes over an uneven road, or a single massive cycle when a car tyre goes into a pothole. For the latter, the vertical acceleration of the passenger can be up to 2g, so the body exerts three times its static force on the seat. The older design of a steel wire spring seat, with a thin layer of foam covering, has low hysteresis. The mass–spring system will vibrate for two or three cycles, whereas motion of a person on an all-foam seat will be overdamped, with no oscillations.

9.3.4 Crash safety The front and rear crumple zones of a car are designed to collapse at a force which transmits a 20g horizontal deceleration to the rigid passenger cage. During a frontal impact, the seat cushion shears because the seat belts do not restrain body motion until their slack is taken up. A lap seat belt can only restrain the pelvis if the body does


217

not sink excessively in the seat. Submarining occurs if the person sinks and moves forwards in the seat, and the seat belt rides up onto the soft organs of the stomach and liver. The use of firm seat foam is ruled out by the requirement in the last section. The problem can be partly solved by providing rigid steel beams at the front of the seat, so that when the body slides forwards, it is forced to rise.

9.4

Foam selection 9.4.1 Foam grades and the indentation force deflection test The specification of open-cell foams for seating will be explored. The foam should have a Young’s modulus of the order of 20 kPa to be comparable with that of human soft tissue. Closed-cell foams have Young’s moduli exceeding 100 kPa, due to the compression of the air trapped in the cells, so are rarely used. The flexible foam industry uses an indentation force–deflection (IFD) test, in which a circular flat indenter of area A ⫽ 323 cm2 (50 in.2) presses on a slab of foam, typically of thickness 100 mm and area 500 mm by 500 mm (ASTM D3574) supported on a flat table. The table is perforated with small holes to allow easy air flow. The foam is preconditioned by compressing it twice to 75% strain, then waiting for 6 min. The force is measured 60 s after a 25% indentation strain is applied, during which time there will be stress relaxation. The 25% IFD result does not necessarily correlate with the seating stiffness for load application times in hours, as the foam can creep. The PFA (www.pfa.org) indicates suitable IFD values (in pounds force for 25% compression) for various applications. In Europe the IFD values are in N. The foam thickness and block size affect the indentation hardness. Therefore the dimensions must be controlled if comparable data is to be generated; it is insufficient to specify minimum values. The foam, surrounding the indenter, lifts from the table (PFA, In Touch, 1994). The top surface bows, and the sides deform inwards, even when the foam block is 500 mm by 500 mm. This affects the indentation load (Table 9.2; see also later). The lowest IFD category super-soft is comparable with polyesterfibre filled cushions. Wolfe (1982) linked the shape of the indentation force–deflection graph to seat comfort, with comfortable foams having more linear graphs. This was quantified by defining a sag or support factor

Sag factor ⬅

65% IFD 25% IFD

(9.2)


IFD range suitable for particular applications 25% IFD (N) 27–53 53–80 80–107 107–133 133–160 160–200 ⬎200

25% IFD (lb force) 6–12 12–18 18–24 24–30 30–36 36–45 ⬎45

Typical use Bed pillows, thick back pillows Back pillows, upholstery padding Thin back pillows, very thick seat cushions Average seat cushions, upholstery padding Firm seat cushions, mattresses Thin seat cushions, firm mattresses Shock absorbing and packaging, carpet pads

A linear force–deflection graph has a sag factor of 2.66; values of 2.8 or higher are regarded as providing good comfort. If the foam has a high initial collapse stress, the sag factor is likely to be low, and the seat may be uncomfortable. The comfort rating of foam seating correlated positively with the rebound resilience of a 21⁄4 in. diameter steel ball of mass 764g, dropped from a constant height (Hilyard, 1984). However foam modulus can vary independently of resilience, while resilience can vary with the ball drop height, hence the maximum foam strain. Dynamic measurements of the foam compressive modulus (Section 2.3) are more relevant to seating design.

9.4.2 FEA of IFD experiments As the IFD test is used to specify foams, its mechanics will be analysed. Mills and Gilchrist (2000) made quarter-scale IFD tests, with an indenter diameter of 50.8 mm, using foam initial thickness T and length (width) L that were multiples of D. Typical data for loading and unloading a PU foam of density 38 kg m⫺3 is shown in Figure 9.7. In the FEA model, the lower flat surface of the indenter is bonded to the upper surface of the foam (in real life there is little sliding at the interface, and the condition makes the FEA more stable). Two vertical mirror symmetry planes, bisecting the foam block, are used to reduce the problem size. The foam was modelled using Ogden strain energy function parameters µ1 ⫽ 20 kPa, µ2 ⫽ 0.2 kPa, α1 ⫽ 20, α2 ⫽ ⫺2, ν ⫽ 0 (Chapter 6). Figure 9.7 shows the predicted variation of the average indentation stress (force/indenter area) with strain. The IFD stress–strain relationship is more linear than the uniaxial stress–strain graph for the foam. The prediction is a good fit to experimental unloading data, being approximately 2 kPa higher for the whole strain range. The experimental loading data is well above the predictions, probably because the model parameters could not reproduce the uniaxial compressive response at low strains (Fig. 6.2).


219

20

Indentation stress (kPa)

PU38

N=2

15 N=1 10

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Average compressive strain under indenter

Figure 9.7

Table 9.3

Predicted indentation stress vs. strain for IFD test with L/D ⫽ 3, T/D ⫽ 1, for the Ogden models N ⫽ 1, µ ⫽ 10 kPa, and α ⫽ 8, and N ⫽ 2, α1 ⫽ 20, µ1 ⫽ 20 kPa, α2 ⫽ ⫺2, and µ2 ⫽ 0.2 kPa, compared with data for loading and unloading PU38 foam (Mills and Gilchrist, 2000).

Peak indentation stress ratios in IFD tests on PU38 foam

T/D

L/D

Hmax (experimental)

0.5 0.5 1 1

2 3 2 3

2.1 2.1 3.0 3.9

Hmax FEA (N ⫽ 2) 1.70 1.75 2.48 2.82

Ratio (experimental/predicted) 1.23 1.20 1.21 1.38

The indentation stress ratio H(ε) was defined in equation (6.14) as the ratio σ–(ε)/σ U(ε) of the mean indentation stress to the uniaxial stress at the same strain. The experimental H, which peaks at ε– ⫽ 0.4, increases with both sample thickness and length (Table 9.3). The predicted H peaks at about 2.8 when the compressive strain is 0.4 (Fig. 9.8). The experimental H data, for loading a PU38 foam cushion of the same shape, has the same shape as, but is higher than, the predicted values. The predicted peak H values for other foam block dimensions (Table 9.3) are about 20% smaller than the experimental values for loading.

220 Polymer Foams Handbook 4 PU38

Indentation stress ratio (H )

3.5

3 N=2 2.5 N=1 2

1.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Average compressive strain under indenter

Figure 9.8

Predicted indentation stress ratio vs. strain for the IFD test of Figure 9.10, compared with loading and unloading data for PU38 foam (Mills and Gilchrist, 2000).

Average indentation stress (kPa) at 25% IFD

6

5 ABAQUS PFA

4

3

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Foam thickness/indenter diameter (T/D)

Figure 9.9

Predicted variation of the 25% IFD stress with foam thickness/indenter diameter (T/D), for L/D ⫽ 3, compared with PU foam data (PFA) for L/D ⫽ 2.5 (Mills and Gilchrist, 2000).

Both Ashe and Grace (1989) and the PFA, In Touch (1994) found that the 25% IFD value increased with foam thickness T. The PFA data, recalculated as the average indentation stress at 25% strain, are plotted in Figure 9.9. Although the L/D ratio for the simulation


10

221

20 40

2

100 8 4 6 Mirror planes (a)

Figure 9.10

(b)

IFD test for foam with L/D ⫽ 2, T/D ⫽ 1, predicted for N ⫽ 2 parameters as in Figure 9.10 at a mean strain ⫽ 0.6: (a) contours of vertical compressive stress σ22 (kPa) and (b) the deformed mesh (initially square) indicates the strain distribution. Tensile stress σ22 contours (kPa) shown on the upper surface (Mills, unpublished).

slightly exceeds the experimental data, and the foams differed, the trends are similar. Therefore, the FEA analysis can successfully predict the effect of foam thickness. PFA data shows that the 25% IFD value increased slightly with the block side L, for a constant foam thickness. A similar prediction is made by FEA (Table 9.3).

9.4.3 Comparison with experimental IFD pressure fields Figure 9.10a shows contours of the compressive stress σ22 in the vertical direction on a deformed IFD specimen with 60% average compressive strain. On the vertical symmetry planes through the foam, the contour for σ22 ⫽ ⫺6 kPa narrows towards the base of the foam, but the ⫺4 kPa contour is nearly vertical. The pressure on the lower foam surface drops gradually with radial distance from the indenter axis. Under the centre of the indenter, there is approximately uniform compression. There are tensile stresses of up to 100 kPa in the foam touching the side of the indenter (Fig. 9.10b); the foam is predicted to wrap around the sides of the indenter, when the imposed strain is high. Davies et al. (2000) observed the pressure distribution using a flexible multi-sensor mat on top of the PU38 foam. The average pressure was calculated as a function of the radial distance from the axis. There was a peak pressure of 8 kPa towards the perimeter of the 101 mm radius indenter, approximately double the 3.5 kPa value at

222 Polymer Foams Handbook the indenter centre, confirming features of the predicted pressure distribution (Fig. 9.13a). The slight lateral load spreading, with increasing distance below the indenter, was also accurately predicted.

9.4.4 Foam selection factors The four main factors in the foam selection for an all-foam car seat are: (a) (b) (c) (d) (e)

cost; low density; vibration isolation; avoiding excessive pressures, either under static or dynamic loading; durability.

Minor increases in materials costs are not tolerated by the automotive industry unless the PU formulation is essential to meet physical property specifications. Natural rubber (NR) foam is much more expensive than PU foam, so is only used in some luxury cars. The foam should have minimum density. Chapter 7 showed that the foam Young’s modulus Ef is related to its relative density R, and the polymer modulus E by Ef ≅ 1.0 ER2

(9.3)

The lowest density foam, that meets the Ef specification, is made from the polymer with the highest E value. The Young’s modulus of semicrystalline PU can be 50 MPa, much higher than the 0.3–10 MPa range for crosslinked NR, which is amorphous. The foam must recover completely from high strain deformation. Chapter 7 shows that the polymer strain in open cell foams is of the same magnitude as the foam compressive strain. Therefore the polymer must behave elastically for strains up to 80%. This excludes polyethylene (PE) and polypropylene (PP) which have yield strains of about 10%. Vibration isolation is important. For a typical resonant frequency f0 ⫽ 4 Hz and an occupant mass m ⫽ 60 kg, equation (9.1) gives a spring constant k ⫽ 40 kN m⫺1. Cavender and Kinklear (1996) loaded a car seat with a mean load of 225 N via an IFD foot, and superimposed a cyclic load at 5 Hz. The mean slope of the force vs. deflection ellipse (Fig. 9.11) indicates a foam dynamic stiffness of about 30 kN m⫺1. However the mean force causes a mean seat compression of about 30 mm, implying a long-time stiffness of 7.5 kN m⫺1, 25% of the dynamic stiffness. The area inside the loop indicates the energy dissipated in each cycle. When the cycling was continued for 17 h, the centre of the ellipse shifted to higher deflection showing that the PU foam crept under the mean force of 225 N.


223

350

Force (N)

300

250

5 min

200

17 h

150

100 20

25

30

35

40

Deflection (mm)

Figure 9.11

Force vs. deflection for an MDI-based PU cushion is cycled (redrawn from Cavender et al., 1996 © SAE).

9.4.5 High resilience PU foams Chapter 2 described the microstructure of high resilience (HR) PU foams. In the USA since 1978, HRII grades of foam used for mattresses must have: (a) density greater than 40 kg m⫺3; (b) ball rebound resilience greater than 60%; (c) sag factor greater than 2.4. HR slabstock foams are made from similar chemicals to HR moulded foams.

9.4.6 Ultra-low resilience PU foams These foams (E-A-R Confor, and Tempur-paedic Temper foams) were originally developed for NASA, for space-flight applications in the 1970s. They are used in ejector seat cushions for military aircraft: under the high strain rates of an ejection the much higher stresses do not cause full compression. If conventional foam were used there could be spinal damage as the seat suddenly bottomed out. They are also used for padding around the cockpit of Formula 1 racing cars, to cushion minor impacts of the driver’s helmet with the cockpit


200

Stress (kPa)

150

100

1 2 3+

50

0

0

0.1

0.2

0.3

0.4

0.5

0.6

Compressive strain

Figure 9.12

Cyclic compression stress–strain curve of Confor 47 foam, for loading at 10 mm min⫺1, labelled with cycle number (Davies and Mills, 1999).

walls. The foams have significantly higher densities (90–100 kg m⫺3) than conventional PU foams for seating, since the long-time Young’s modulus of the PU is much lower than for conventional seating foam. The higher density means that the strains in the PU are higher than usual, at a certain foam compression (Chapter 7). This will increase the non-linearity of the viscoelastic response. These foams, with glass transition temperatures close to 20°C (Davies and Mills, 1999), are advertised for use in wheelchair seating and mattresses. Small samples feel completely different to conventional foam, due to the viscoelastic creep and recovery on the 1–30 s timescale, and indentations remaining for a number of seconds. While these phenomena are attention grabbing, they are not necessarily related to seating or sleep comfort. When Confor or Tempur foams are loaded and unloaded (Fig. 9.12), a large fraction of the energy input is absorbed as heat. The major cause of the hysteresis is the proximity of the glass transition temperature of the polymer, hence the high tan δ value. The restricted flow, of air through holes in the cell faces, only contributes to energy losses at impact deformation rates. Davies and Mills (1999) compared the compressive creep and seating pressure distributions of Confor type foams and slabstock PU foams, on a timescale of 10 min or longer, and found relatively minor differences. The characteristic relaxation time of the foam is much less than 10 min, so at longer times its response is nearly elastic. Since its Young’s modulus and indentation resistance are similar to those of conventional foams, it is not surprising that the pressure distributions


225

are similar. Whether or not Confor-type foam mattress overlays improve sleep remains to be determined; there is no conclusive evidence so far. Indentations under the body, which persist for a few seconds, may affect the frequency of turning in the night.

9.5

Seat design 9.5.1 Uniform uniaxial compression This and the following sections deal with the mechanics of a person sitting on a flat slab of foam supported on a rigid flat base (an approximation for the static loading of an all-foam car seat), using increasingly complex models. The simplest model takes the buttocks as a rigid flat surface of area equal to the foam cushion area. It provides order of magnitude estimates of the foam Young’s modulus and cushion thickness, by ignoring the non-linearity and rate dependence of the cushion response. The buttock area A is approximately 0.3 m by 0.3 m. The spring constant k of a uniformly compressed foam block is related, using the definitions of uniform stress σ and strain ε, to the block thickness t and the foam (secant) Young’s modulus Ef by k ≡

EA F σA ⫽ ⫽ f x εt t

(9.4)

The quantities Ef and t are determined by a condition for the resonant frequency, and a condition that the seat should not bottom out for a passenger vertical acceleration of 2.5g; a peak force on the seat of 3.5 ⫻ 600 ⫽ 2 kN. At this force, assuming a linear response, the deflection x ⫽ t, so F ⫽ EfA. Using A ⫽ 0.09 m2, Ef ⫽ 22 kPa. From page 222, the resonant frequency condition leads to k ⫽ 40 kN m⫺1. Hence, by equation (9.3), t ⫽ 0.05 m. If the PU has a Young’s modulus E ⫽ 40 MPa, to obtain a foam modulus of 22 kPa requires a relative density of 0.025 (by equation (7.3) with C1 ⫽ 1.0). Since the solid PU has the density 1200 kg m⫺3, the PU foam has the density 30 kg m⫺3.

9.5.2 Indentation with a rigid butt-form Shen and Vertiz (1997) measured the force indentation relationship (Fig. 9.13) when a rigid butt-form indenter was used with three US auto seats. However the shape of a subject’s buttocks depends on the seat stiffness and changes under load. Consequently, the wooden butt-form shape was a poor approximation for part of the load range.

226 Polymer Foams Handbook 1.4 1.2

Force (kN)

1 0.8 0.6 1

0.4

2 3

0.2 0

Figure 9.13

0

10

20

30 40 50 Deflection (mm)

60

70

80

Force–deflection graphs for three US car seats (redrawn from Shen and Vertiz, 1997 © SAE).

They found a typical deflection of 50 mm for a 700 N load, which means an average (creep) stiffness of 14 kN m⫺1. In the initial stages of compression, the seat stiffness is lower. The seat should not bottom out for vertical accelerations of the passenger of 2.5g; a peak force on the seat of 3.5 ⫻ 600 ⫽ 2 kN. As the seat force–distance relationship has positive curvature, this condition can be met with a total seat thickness of about 120 mm. The force–indentation relations were more linear than the foam compressive stress–strain curve.

9.5.3 Indentation with a compliant dummy A manikin for wheelchair cushion testing is under development by ISO Technical Committee 157 (2001). The SKELI device consists of the bones of the pelvis, lower spine, and upper legs, plus a PU gel moulding that simulates the soft tissue. The external shape of the buttock moulding was probably taken from a sitting person, but its relation to the average buttock dimensions of the at-risk population is unknown. A standard load (830 N) and standard soft tissue geometry and properties are needed for comparisons between seats at different laboratories to be reproducible. Human subjects have been used to compare a small range of cushions, but the pressure distributions cannot be compared with measurements in other laboratories, and there is debate about the appropriateness of the subjects chosen. While the SKELI dummy is an improvement over a rigid butt-form,


Muscle

227

Bone Buttock model

Fat Skin

Cushion

Figure 9.14

2.0 (mPa) 1.6 1.2 0.8 0.4 2.0

2-D computer model of a deformed buttock of a 80 kg male on a foam cushion, with contours of von Mises stress (Bouten et al., 2003, © American Congress of Rehabilitation Medicine).

the shape of the gel moulding is a compromise, and the gel properties may not match those of the average human soft tissue.

9.5.4 FEA of buttock and foam deformation Early FEA of the buttocks on a foam cushion used 2-D models (Dahnichki et al., 1994), so could not simulate important aspects of the geometry. Bouten et al. (2003) used such a model, with the cushion, and skin, fat, and muscle layers to show (Fig. 9.14) that there were von Mises stress (maximum shear stress) maxima at locations in the subcutaneous fat and muscle layers, where pressure sores can occur. However, no details of the material parameters were given, and the stress units, given as mPa, are probably 10 kPa. Verver et al. (2004) used a 3-D model (Fig. 9.15). The predicted principal compressive stresses are 60 kPa close to the ischial tuberosities, but only 20 kPa at the buttock/cushion interface. The foam was modelled as a linearly elastic material with a Young’s modulus E ⫽ 200 kPa, while the soft tissue was modelled as a Mooney Rivlin solid with C1 ⫽ 1.65 kPa, C2 ⫽ 3.35, and ν ⫽ 0.49. The higher stresses inside the buttock tissue indicate the probable site of pressure sore formation. Hence surface pressure measurements on the buttocks may not reveal the stresses that cause pressure sores. The predicted pressure distribution on the cushion surface (Fig. 9.15b) has the main features of the experimental patterns (Fig. 9.3). However, the foam would be better modelled as an Ogden hyperfoam material.

228 Polymer Foams Handbook 25 20 15 10 5 0 (a)

(b)

Figure 9.15

9.6

(a) 3-D buttock model sitting on a foam cushion and (b) predicted interface pressure map on the cushion surface, with pressure key in kPa (Verver et al., 2004 © Taylor & Francis).

Other foam mechanical properties The following sections consider foam property requirements, other than elastic, that must be met in a successful seating product.

9.6.1 Mechanical fatigue The durability of mattresses and car seats is important, since consumers expect the performance to be undiminished after perhaps 10 years. Durability tests attempt to accelerate deterioration processes. Mechanical fatigue uses high repeated loads; a cylindrical roller can be reciprocated horizontally across the product, and the height loss measured after a 75,000 cycles. This simulates a high load applied 40 times a night for 10 years. Foam height loss and IFD loss are larger for low foam densities. The maximum foam compressive strain in fatigue tests affects the amount of cell damage, the fracture of foam cell edges, and permanent softening of the polymer in the foam. Low density foams cannot be used in car seats, because of their inferior fatigue properties. The type of PU foam is matched to the area of use in a car (Casati et al., 1999). High performance foams are used for the front seat cushions, whereas lower density foams are used for seat backs and rear bench seats. Blair and Horn (1998) investigated the durability of PU seats in police fleet use. They found that TDI-based PU foams gave excellent performance for at least one vehicle lifetime. Cushions made with MDI, or containing some recycled polyol, did not perform quite as well.

Chapter 9 Seating case study 5

7.5

10

12.5

15

100

17.5

90 17.5

80 70

15

Hot cure 33 kg/m3

60 50

12.5 10

40

7.5

5

Relative humidity (%)


100

30

40

60

80

90 80

80 70

MDI HR 50 kg/m3

60 50

60 40 20

40 30

100

50 60 70 Temperature (°C) 25

50

80

75

100 100

90 80

75

70

TDI HR 32 kg/m3

60

50

50 25

40

40

100 Relative humidity (%)

40


20

229

50 60 70 Temperature (°C) 20

40

80

60

80

90 80

80

70

60

TDI HR 50 kg/m3

60 50

40 20

40 30

30 40

Figure 9.16

50 60 70 Temperature (°C)

80

40

50 60 70 Temperature (°C)

80

Contours of per cent wet compression set on temperature–humidity graphs for four types of foam (Broos et al., 2001 © Rapra).

9.6.2 Hydrolysis PU foams can react rapidly to atmospheric moisture levels, since there is a high surface area, a low diffusion distance into the foam cell edges, and the equilibrium water absorption is significant. Nevertheless, the timescale for reaching equilibrium water content is large. For some foams, water absorption affects the Tg of the polymer and therefore the creep of the foam. The deterioration of PU automotive seats was studied by Broos et al. (2001), using accelerated durability tests, under extreme conditions, to rank the performance of various PU foams. The compression set tests, at up to 80°C and 95% relative humidity, lasted for 22 h. The compression set increases with temperature and humidity (Fig. 9.16). The foams which performed best were hot cure (high resiliency foams using 80% TDI with a polymeric diphenyl methane diisocyanate), while TDI foams of density 50 kg m⫺3 were more resistant to compression set than those of 32 kg m⫺3 density. It is hoped that the same ranking applies under more normal conditions. The type of PU used should be matched to the weather conditions in the country of use.

230 Polymer Foams Handbook 9.6.3 Additives to provide fire resistance Fire regulations for domestic furniture often lead to the wrapping of PU foam with woollen or other non-woven layers, so the risk of a smouldering cigarette causing a fire is reduced. A variety of solid additives, such as aluminium hydroxide, antimony chloride, or melamine, can be added to PU foams to suppress flame or to improve the fire resistance. These additives modify the mechanical properties of the foam, since they have higher moduli, and often have weak interfaces with the PU.

Summary The design of car or wheelchair seating is only approximate, since a complete computer model, with detailed anatomical and foam geometry, is not feasible at present. This is due to inadequate materials models for soft tissue, which have parameters that change with muscle tension. However, given the weight and size variability among the population, an ideal seat for one person would be suboptimal for many others. Car seat preferences also differ from country to country. The seat design procedures of Section 9.5 determine the main parameters. A foam seat must be thick enough not to bottom out, and the foam compressive modulus must be of the same order as that of soft tissue. Slab foam seating, widely used in domestic furniture, does not provide an ideal pressure distribution. However, even complex multimaterial seats do not produce a uniform pressure distribution. The most important material selection criterion may well be durability in moist environments. The chemistry of the PU foam process can be adapted to give durable foams, and the moulding process provides low cost manufacture. Chapter 7 explored the microstructural reasons for the high elastic deformation of these materials.

References Aissaoui R. et al. (2001) Analysis of pressure distribution at the body-seat interface, Med. Eng. Phys. 23, 359–367. Andreoni G., Santambrogio G.C. et al. (2002) Method for the analysis of posture and interface pressure of car drivers, Appl. Ergon. 33, 511–522. Ashe W.A. & Grace O.M. (1989) Polyurethanes tomorrow – testing of slab foam, J. Cell. Plast. 25, 371–379. Bader G.B. & Engdal S. (2000) The influence of bed firmness on sleep quality, Appl. Ergonom. 31, 487–497.


231

Blair G.R. & Horn R.J. (1998) Fleet durability testing of moulded PU foam and competitive automotive cushions, Woodbridge group.com. Blair G.R., Milivojevich A. et al. (1999) Automotive seating comfort; defining comfort properties in PU foam, SAE Int. Congress Detroit, Paper 587. Bouten C.V., Oomens C.W.J. et al. (2003) The etiology of pressure ulcers: skin deep or muscle bound? Arch. Phys. Med. Rehabil. 84, 616–619. Brienza R.P. & Geyer M.J. (2001) Understanding support surface technologies, Adv. Skin Wound Care, on www.Woundcarenet.com in Clinical Management Extra. Brienza R.P., Karg P.E. & Brubaker C.E. (1996) Seat cushion design for elderly wheelchair users based on minimising of soft tissue deformation using stiffness and pressure measurements, IEEE Trans. Rehabil. Eng. 4, 320–327. Broos R., Herrington R.M. & Casata F.M. (2001) Endurance of PU automotive seating foam under various temperature and heating conditions, Cell. Polym. 19, 169–204. Casati F.M., Berthevas P.R. & Herrington R.M. (1999) The contribution of molded PU foam characteristics to comfort and durability of car seats, SAE Trans. J. Mater. Manuf. 108, 493–503. Cavender K.D. & Kinklear M.R. (1996) Real time dynamic comfort and performance factors of polyurethane foam in automotive seating, in Automotive Design Advancements in Human Factors, SAE SP-1155, pp. 53–68. Dahnichki P.A., Crocombe A.D. & Hughes S.C. (1994) Deformation and stress analysis of supported buttock contact, Proc. Inst. Mech. Engr. Part H, J. Eng. Med. 208, 9–17. Davies O.L. & Mills N.J. (1999) The rate dependence of Confor PU foams, Cell. Polym. 18, 117–136. Davies O.L., Gilchrist A. & Mills N.J. (2000) Seating pressure distributions using slow recovery PU foams, Cell. Polym. 19, 1–24. Ferguson-Pell M.W. (1990) Seat cushion selection, J. Rehabil. Res. Dev. Clin. (Suppl. 2), 49–73. FSA, Vista Medical at www.verge.com. Fung W. & Parsons K.C. (1996) Some investigations into the relationship between car seat cover materials and thermal comfort using human subjects, J. Coat. Fabric. 6, 147–176. Goosens R.H.M. & Zegers R. (1994) Influence of shear on skin oxygen potential, Clin. Physiol. 14, 111. Griffin M.J. (1990) Handbook of Human Vibration, Academic Press, London. Gurram R. & Vertiz A.M. (1997) The role of automotive seat cushion deflection is improving ride comfort, in Progress in Human Factors in Automotive Design, SAE SP-1242, pp. 153–159.

232 Polymer Foams Handbook Hubbard R.P. et al. (1993) New biomechanical models for automotive seat design, Seat System Comfort and Safety, SAE SP-963, pp. 35–42. ISO TC 173/SC 1/WG 11 (2001) Wheelchair seating. Part 2. Test methods for devices intended to manage tissue integrity – seat cushions, available at www.wheelchairstandards.pitt.edu. Kinkelaar M.R., Cavender K.D. & Crocco G. (1998) Vibrational characterization of various polyurethane foams employed in automotive seating applications, J. Cell. Plast. 34, 155–173. Lueder R., Ed. (1994) Hard Facts About Soft Machines, Taylor and Francis, London. Milivojevich A., Stanciu A. et al. (2000) Investigating psychometric and body pressure distribution responses to automotive seating comfort, Human Factors in 2000, SAE World Congress, Paper 626. Mills N.J. & Gilchrist A. (2000) Modelling the indentation of low density polymer foams, Cell. Polym. 19, 389–412. Patten W.N., Sha S., & Mo C. (1998) A vibration model of open celled polyurethane foam automotive seat cushions, J. Sound Vib. 217, 145–161. Polyurethane Foam Association www.pfa.org/intouch/index/html. PFA (Polyurethane Foam Association) web site www.pfa.org contains articles from the magazine In Touch. PFA (1994) How foam firmness affects performance, In Touch 4(3). Podoloff R.M. (1993) Automotive seating analysis using thin flexible tactile sensor arrays, Seat System Comfort and Safety, SAE SP-963, pp. 59–65. Pywell J.F. (1996) Automotive seat design affecting comfort and safety, Seat System Comfort and Safety, SAE SP-963, pp. 13–24. Setyabudhy R.F., Ali A. et al. (1997) Measuring and modelling of human soft tissue and seat interaction, Progress with Human Factors in Automotive Design, SAE SP-1242, pp. 135–142. Shen W. & Vertiz A.M. (1997) Redefining seat comfort, Progress in Human Factors in Automotive Design, SAE SP-1242, pp. 161–168. Stabler K.M., Cullen E. et al. (1996) How people sit in cars, IRCOBI Conference, IRCOBI Secretariat, Bron, France, pp. 321–338. Stekelenburg A., Oomens C.W.J. et al. (2006) Compression induced deep tissue injury examined with magnetic resonance imaging and histology, J. Appl. Physiol. 100, 1946–1954. Tan A., Deno L. & Leenslag J.W. (1996) Low density all-MDI polyurethane foams for automotive seating, Cell. Polym. 15, 250–265. Tekscan brochures (2000) at www.tekscan.com. Vakeva R. (1997) A good night’s sleep, A polyurethane art, Polyurethane World Congress, 242–250.


233

Verver M.M., van Hoof J. et al. (2004) A FE model of the human buttocks for prediction of seat pressure distributions, Comp. Meth. Biomech. Biomed. Eng. 7, 193. Wolfe H.W. (1982) Cushioning and Fatigue, Chapter 3, in Mechanics of Cellular Plastics, Ed. Hilyard N.C., MacMillan, London.

Chapter 10

Sport mat case study

Chapter contents 10.1 Introduction 10.2 Modelling of impacts 10.3 Experimental impacts 10.4 Fall mat design 10.5 Martial arts mats References

236 237 243 246 246 249


10.1

Introduction 10.1.1 Mats used in sport Crash mats are used in a number of sporting applications, for example below indoor climbing walls, and as the landing area in pole vaulting. After a fatality at a dry ski slope, where an inexperienced skier lost control and hit the barrier at the end of a short slope, the Health and Safety Executive recommended that a thick foam mat barrier was installed. Lyn and Mills (2001) considered the design of such mats. In the European Standard (BS EN 12503: 2001) for sports mats, there are a range of mat types. The thicker mats for high jump and pole vault are impact tested with a 30 kg mass, having a 75 mm diameter hemispherical end, falling vertically though 1.2 m. The peak acceleration must be less than 10 g. Neither the link between the striker shape and mass and the human body nor that between the peak acceleration and injury mechanism are explained. The working documents of the EN standards committee are not available to the public, but the test criteria should be linked to injury mechanisms or sport requirements. The test methods should consider how soft tissue deformation changes the effective mass (Section 17.3.2) of the torso. Martial arts mats must both protect the players against fall injuries and also allow foot movements without the risk of ankle sprains. The upper and lower surfaces of the mat must provide a high friction coefficient, so the players do not slip and the mat remains in a stable position on the floor. For all these applications, the performance of the mat must be stable over a long period, so the foam impact response must not change with impact number. There is related research on the padding of rugby goal posts with cylindrical pieces of polyurethane (PU) foams (Mills and Lyn, 2001).

10.1.2 Foam materials Most fall mats use chip foam, described in Chapter 2, covered with waterproof fabric-reinforced polyvinylchloride (PVC). Mats of thickness 0.1, 0.2, and 0.4 m, with width and length of 1 m, were tested by Lyn and Mills (2001). Table 10.1 gives the densities and the relative compressive stress at a 0.3 strain, measured for compression through the mat thickness. The colour mix of the 0.2 m thick foam differed from the other mats, and it has a 30% higher compressive modulus. The foam nominal density was 4 lb ft⫺3 (64 kg m⫺3).

10.1.3 Head impacts A worse-case scenario was considered, when the head hits the mat before other body parts. If the arms, legs, or torso hit the mat first,

Chapter 10 Sport mat case study

Table 10.1

237

PU remoulded foams examined Mat thickness (m) 0.1 0.2 0.4

Foam density (kg m⫺3)

Relative compressive stress at ε ⫽ 0.3

63.3 75.7 ⫾ 0.7 72.2 ⫾ 1.0

1.00 1.29 1.00

the head accelerations will be smaller. Although the body mass (50–100 kg) is much greater than the head mass (4–6 kg), the flexibility of the human neck allows the effective separation, on the short timescale of the head impact, of the head and neck motion. Even in the unlikely event of the head and body velocities being in line, and the impact being on the crown of the head, the neck will buckle. Hulme and Mills (1996) showed, using dummies with realistically flexible necks, that only the head mass m was involved in the dynamics of the head’s motion, so the force F on the head was related to its acceleration a by Newton’s second law F ⫽ ma

(10.1)

The head impact has a kinetic energy KE given by KE ⫽

1 2

mV 2

(10.2)

where V is the head velocity. It is necessary to know the maximum velocity at which a falling athlete, or out-of-control skier, could impact the foam mat. An upper limit on the skier’s head velocity on a practice slope is 10 m s⫺1. The velocity of the falling climber depends on the fall distance, but as climbing walls are rarely ⬎5 m high, the head velocity is unlikely to be ⬎10 m s⫺1. If the head velocity is 10 m s⫺1, then the head kinetic energy is 250 J, and at 5 m s⫺1, it is 62.5 J. Chapter 14 explains the risk of concussion or minor brain injury when the force on the head exceeds 5 kN (the headform acceleration exceeds 100 g).

10.2

Modelling of impacts 10.2.1 Type of analysis The large change in safety mat geometry during a head impact requires the use of finite element analysis (FEA). The foams are non-linear,

238 Polymer Foams Handbook viscoelastic materials that are porous to air, so approximations can be made. We will consider separately: 1. Static FEA, assuming the foam is a non-linear elastic material, validated by comparison with experiments (Lyn and Mills, 2001). 2. Dynamic FEA, assuming the foam is a non-linear viscoelastic material (Mills, 2006). The models ignore cross-effects, such as increased air pressure stabilising the cell edges against bending, increasing the foam compressive resistance.

10.2.2 Hyperfoam model parameters for FEA The foam characterisation was at impact rates, in a test lasting 0.05 s. The compressive stress–strain curve of the rebonded PU foam was measured using a 100 mm cube sample, taken from the 0.1 m thick mat, impacted from heights of 0.005–0.65 m by a flat anvil of mass 1.50 kg. The latter supported an accelerometer and fell without friction on two guide wires. Analysis in Chapter 8 showed that, for a 100 mm cube impacted at 3 m s⫺1, the contribution of air flow is negligible. The stress–strain curve (Fig. 10.1) shows a near-linear initial portion and a steeply rising 50

Stress (kPa)

40

30

20

Model 200 mm

10

0

100 mm

0

10

20

30

40

50

60

70

80


Figure 10.1

Impact compression stress–strain response for 0.6 m drop on 100 mm cube of remoulded PU foam, and 200 ⫻ 200 ⫻ 100 mm specimen, compared with model prediction, parameters in Table 10.2 (Lyn and Mills, 2001).

239


region for strains ⬎60% due to cell edges making contact. For the 200 mm by 200 mm sample, the stress plateau of about 10 kPa between strains of 10% and 20% is likely to be due to a wave of densification, modelled by dynamic FEA in Chapter 6. Hence the underlying response was taken to be a linear increase in stress. The foam Poisson’s ratio was measured both in compression and tension, using apparatus described in Chapter 4, at moderate strain rates on an Instron machine. The Poisson’s ratio in compression was small, as for virgin PU flexible foam, but in tension the value increased until, it was 0.44 for tensile strains ⬎0.1. This is an evidence for a change in deformation mechanism between compression and tension. Figure 10.2a shows the tensile response and Figure 10.2b the variation of the lateral strain with the tensile strain. The parameters of the hyperfoam model (Chapter 6) were chosen (Table 10.2) to fit the experimental data in Figures 10.1 and 10.2. The term α ⫽ 2 may represent the rubber-like elasticity of extending foam cell edges, whereas the term α ⫽ 8 mainly models the touching of foam edges at high strains.

10.2.3 Static FEA model The headform was modelled as a sphere of radius 100 mm (Fig. 10.3) made of aluminium with Young’s modulus 70 MPa, so was effectively 25

0

Model

−2 Lateral strain (%)

Stress (kPa)

20

15

10

−4 Model

−6 −5

5

−10

0 0

5

10

(a)

Figure 10.2

Table 10.2

15

20

25

30

0

5

10

(b)

Strain (%)

15

20

25

30

Tensile strain (%)

(a) Tensile stress–strain response of remoulded PU foam of density 63 kg m⫺3, at strain rate 0.3 min⫺1. (b) Lateral strain vs. tensile strain, compared with model of Table 10.2 (Lyn and Mills, 2001). Parameters for the hyperelastic model of rebonded PU foam N

Shear modulus µ (kPa)

Exponent α

Poisson’s ratio ν

1 2

18 1.2

8 ⫺2

0 0.45

240 Polymer Foams Handbook Rotational symmetry axis

Metal headform

70 90 50 30

Foam

10

Rigid support table

(a)

Figure 10.3

(b)

(a) Predicted deformed shape of 400 mm thick foam mat after 207 mm vertical deflection by a 100 mm radius headform. (b) Predicted contours of principal compressive stress (kPa) (Lyn and Mills, 2001).

rigid compared with the foam. The foam was supported by a rigid flat table, and a coefficient of friction of 0.75 used between the foam and both the headform and the table. To reduce the computation time, an axisymmetric problem was considered, with the foam being a vertical cylinder of radius 500 mm. The elements used were of CAX8 type, four-node biquadratic axisymmetric quadrilaterals in ABAQUS standard. The FEA mesh for the undeformed foam consisted of horizontal and vertical straight lines, with spacing biased towards the initial impact point. For small foam deflections, the contact area with the headform increases as the indentation increases. After 207 mm vertical deflection by the headform (Fig. 10.3a) the foam upper surface contacts the lower half of the headform surface, but its mesh spacing has hardly changed at the lower and right-hand sides. There are high compressive strains ε2, and small tensile strains ε1, in the foam just below the headform near the vertical axis. The distorted angles at the mesh intersections revealed shear strains in the foam at the sides of the headform. In Figure 10.3b, the contours of compressive principal stress are closely bunched under the vertical axis of the indenting headform. The maximum compressive stress of 100 kPa is low compared with values exceeding 500 kPa in the foams of protective helmets. The predicted head force vs. mat deflection graphs (Fig. 10.8) have positive curvatures. The model can predict the peak headform force as a function of the kinetic energy (Fig. 10.4). Since the model is elastic, the energy stored by the foam is equal to the area under the force–deflection graph. If 100g head acceleration is allowable (a force of 5 kN), a 200 mm thick mat can absorb 140 J safely, while a 400 mm thick mat can absorb 450 J safely. If 200g head acceleration is tolerated, the safe


241

10

8

Force (kN)

0.4 m 0.2 m 6

4

2

0

0

100

200

300

400

500

600

700

800

Input energy (J)

Figure 10.4

Predicted maximum impact force on 100 mm radius headform vs. impact energy for PU chip foam mats of nominal density 64 kg m⫺3 of the thickness indicated (Lyn and Mills, 2001).

impact head energies will be much greater. Therefore a 400 mm thick mat, containing PU chip foam of 64 kg m⫺3 density, can prevent head injuries for impact velocities ⬍10 m s⫺1.

10.2.4 Viscoelastic FEA model Mills (2006) used a linear viscoelastic model with a hyperelastic foam in ABAQUS Explicit to model head impacts on mats. The shear relaxation moduli µ for the PU foam (Table 10.1) were increased by 20% (to maintain the values at a time of 10 ms) then used as the instantaneous shear modulus G0 in n ⎛ ⎞⎟ G(t) ⫽ G0 ⎜⎜⎜1 ⫺ ∑ gi exp(⫺t /τ i )⎟⎟⎟ ⎜⎝ ⎟⎠ i ⫽1

(10.3)

where gi is the dimensionless shear relaxation modulus associated with relaxation time τi. Values of gi ⫽ 0.1 were used for τi ⫽ 1, 3, 10, 30, 100, 300, and 1000 ms. The sum of the gi cannot exceed 1.0 since the long-term modulus G⬁ defined by n ⎛ ⎞⎟ G⬁ ⫽ G0 ⎜⎜⎜1 ⫺ ∑ gi ⎟⎟⎟ ⎜⎝ ⎠ i ⫽1 ⎟

must be positive.

(10.4)

242 Polymer Foams Handbook Head

0.5 0.25

Mat

Symmetry axis

0.25 0.5 Rigid table

Figure 10.5

Dynamic FEA prediction 102 ms from the start of head impact on a 400 mm mat showing the principal compressive stress contours (kPa) just after head rebound (Mills, 2006).

1.2

0.6

1.0

0.4 m mat

0.1 m mat 0.5

0.8 0.6 0.25

0.4

0.4

0.5

0.3

0.25

0.2 0.1

0.2 0

1.0

0.5 Head force (kN)

Head force (kN)

1

0

(a)

Figure 10.6

20

40

60

Mat deflection (mm)

0

80 (b)

0

20

40

60

80

100 120 140

Mat deflection (mm)

Predicted headform force vs. central mat deflection graphs for the marked drop heights (m) for mats of thickness (a) 0.1 m and (b) 0.4 m (Mills, 2006).

The foam mat, a cylindrical block of radius 600 mm and thickness 100 or 400 mm, was impacted by a falling rigid headform of mass 4.1 kg and radius 87 mm (the mean radius of curvature of the headform at the crown impact site used in experiments). The boundary conditions were as before (Fig. 10.3). Figure 10.5 shows the headform just after rebound from the mat; there are still residual stresses as the viscoelastic mat recovers. The predicted headform force (head acceleration multiplied by its mass) vs. the central mat deflection graphs in Figure 10.6 have a


Table 10.3

243

Energy flows and hysteresis predicted for head impacts on foam blocks Mat thickness (m) 0.1 0.1 0.1 0.4 0.4 0.4

Drop height (m)

Maximum deflection (mm)

Energy input Ein ( J)

Hysteresis (%)

0.25 0.5 1.0 0.25 0.5 1.0

63.9 77.7 88.8 77.0 101.0 131.3

10.0 20.1 40.3 10.0 20.1 40.2

35 35 35 39 40 42

hysteresis that is almost constant when the headform drop distance (hence initial impact velocity) is varied. The mat hysteresis is defined as the energy lost in a loading/unloading cycle divided by the kinetic energy of the impact. The energy inputs and losses are given in Table 10.3. For the 400 mm thick mat, the predicted loading response is approximately the same as the experimental data in Figure 10.8. The viscoelastic model is more successful than the purely hyperelastic foam model, shown by the dashed curves in Figure 10.8. For the 100 mm thick mat, a head impact velocity of 4.43 m s⫺1 (a drop from 1.0 m) causes the foam to almost bottom out. As the impact velocity (drop height) increases, the experimental graphs show increased hysteresis, and the unloading curves fall below the predicted responses. Hence linear viscoelasticity cannot completely explain the foam mat response. Non-linear viscoelasticity in the foam is the most likely cause of the extra hysteresis. There is possibly a contribution from air flow through the foam (Chapter 8). However, at present, FEA cannot model such a contribution.

10.3

Experimental impacts 10.3.1 Falling headform test The aluminium headform, of circumference 58 cm, had mass 4.1 kg. Its shape, defined in BS EN 960: 1995, had a 100 mm radius of curvature for the fore and aft section, and 75 mm for the ear-to-ear section, at the crown impact site. It was fitted with a linear accelerometer aligned vertically (Fig. 10.7), and dropped from heights from 0.125 to 1.0 m. The deceleration data was converted (as explained in Chapter 5) into a graph of the force on the headform vs. the deflection of the upper surface of the foam at the point of impact (Fig. 10.8). The nearly linear loading parts of the curves almost follow a ‘mastercurve’ when the deflection is increasing. The loading curves for the higher impact velocities lie slightly above those for lower velocities indicating some


Cable

30 V AC Electromagnet Accelerometer Coupler

Headform

Drop height

Foam mat

Concrete floor

Free fall of a headform instrumented with a linear accelerometer.

1.2

Model

Head force (kN)

1

0.75 m

0.8 0.50 m

0.6 0.25 m

0.4 0.2 0

0.8

1m

Head force (kN)

Figure 10.7

0.6 0.48 m

0.4 0.25 m

1m 0.75 m

0.125 m

0.2

0.125 m

0

(a)

Figure 10.8

20 40 60 Mat deflection (mm)

0

80 (b)

0

20

40 60 80 100 Mat deflection (mm)

120

Headform impact tests, onto remoulded PU foam mats of thickness (a) 0.4 m and (b) 0.1 m, for drop heights marked compared with hyperelastic model predictions for 100 mm radius headform (Lyn and Mills, 2001).

rate dependence in the response. When the headform velocity approaches zero, the force drops below the mastercurve, and during unloading the force is far below the loading value. The initial loading stiffness is almost independent of the mat thickness, being 10, 9 and 7 N mm⫺1 for the 100, 200, and 400 mm thick


245

Maximum deceleration (g)

30

25

20

15 0.4 m 0.2 m 0.1 m

10

5

1

2

3

4

5

Velocity of 5 kg head (m/s)

Figure 10.9

Maximum experimental deceleration of 4.1 kg headform vs. impact velocity for three mat thicknesses (Lyn and Mills, 2001).

mats, respectively. However, if the strain exceeds 60%, the foam begins to bottom out, and the slope of the loading curve becomes steeper (Figure 10.8b).

10.3.2 Effect of headform velocity The maximum headform deceleration amax is related to the peak force Fmax on the headform by Newton’s second law (equation (10.1)). At the maximum foam deflection xmax, the impacting headform has momentarily stopped, so an energy equal to the initial KE has been transferred to the foam. The peak headform accelerations were plotted against the impact velocity V (Fig. 10.9). For the thicker mats, there is a linear relationship with equations 0.4 m thick

amax ⫽ 3.62V ⫹ 1.45

0.2 m thick

amax ⫽ 5.37V ⫺ 0.94

(10.5)

For the 0.1 m thick mat, the relationship becomes non-linear for the higher impact velocities, when the peak foam strains ⬎70%. The foam performance does not deteriorate with repeated impacts; when the headform was dropped 5 times, at 10-min intervals, from 1.0 m to 0.4 m thick mat, there was only a ⫾5% variation in the maximum deceleration, with no discernible trend. When the cover of the 0.4 m thick mat was removed, and impact tests repeated, the

246 Polymer Foams Handbook maximum deceleration increased by ⬍1 g. Hence the foam provides all the resistance to indentation. The predicted headform forces (Figure 10.6) are 33% smaller than the experimental loading forces for the 400 mm mat. It would be necessary to remodel the tensile responses of Figure 10.2 to ensure that the viscoelastic parameters are realistic. Nevertheless the viscoelastic model is superior to the hyperelastic one.

10.4

Fall mat design The modelling shows that PU chip foam mats are non-linear materials, with a non-linear viscoelastic response. The air-flow contribution to the response is likely to be minor under most impact conditions. For the high jump or pole vault, a thick mat reduces the fall distance, hence the athlete’s kinetic energy at impact. However, the foam mat for some applications could be thinner if the PU chip foam had a higher modulus or if it was replaced by another material. Generalising from helmet design, the ideal foam should just absorb the design impact energy without injury. For the current foam fall mats, the headform force is usually almost linearly proportional to the foam deflection. As the impact kinetic energy increases, the mat thickness must also increase, and the foam Young’s modulus must decrease. For a minimum thickness mat, the force on the head would ideally be nearly constant at a level just below that needed to cause concussion. However no current designs have this type of response.

10.5

Martial arts mats Other foam mats, such as martial arts mats, also have the requirement that the player must move around safely on the mat. The player’s feet must not excessively indent the mat, otherwise there might be ankle injuries. The mats must also protect against head impacts, on the rare occasions when uncontrolled falls occur. Although denser versions of chip foam can be used for these mats, the design requirements are closer to those for playground surfaces than those for crash mats. Some mats for martial arts are made from open-cell polymeric foams while others are made from closed-cell foams (Table 10.4). The upper surface tends to be textured PVC, bonded to the foam. The closed-cell polymer foams have Young’s moduli ⭓ 100 kPa due to the air content, while open-cell flexible PU foams can have moduli in the range 10–300 kPa. Mills and Lyn (2004) designed a test rig to simulate an oblique impact of a foot on a mat. The rig is a compromise between simplicity,


Table 10.4

247

Foam types used in martial arts mats Typical density (kg m⫺3)

Polymer

Foam structure

Isotropic

Rebonded polyurethane Polyethylene Vinyl nitrile

Open cell

No

100–250

Closed cell Closed cell

Nearly Nearly

40–150 Circa 100

Track

V

Carriage Springs Foot Mat V

F

Load cells Low friction support

Figure 10.10 Schematic of test rig for oblique foot impacts (Mills and Lyn, 2004). ruggedness, and biofidelity. A rigid metal foot was used in place of multiple bones and muscles. It had a single axis of ankle rotation, with closed-cell foam springs providing the non-linear stiffness. The axis of rotation was horizontal rather than the 45° upwards axis of the subtalar joint when the foot plantar surface is on the ground. The ankle was an adaptation of a neck design (Gilchrist and Mills, 1996), in which metal parts limit rotation to 30°. The initial stiffness was 4 Nm for 15°, with an increase by a factor of 5 above that angle (Wright et al., 2000). The foot rotation was monitored using a rotational potentiometer. The mat was mounted on a metal plate, supported by two quartz tri-axial force cells (Kistler 9348B). To achieve an oblique impact, the 5 kg foot and ankle fell vertically on a rail through 0.6 m, while a 6.9 kg ‘floor’ and mat were accelerated horizontally by a pneumatic cylinder to a velocity of 2 m s⫺1 pre-impact, then was free to decelerate (Fig. 10.10). When the foot and mat met, the deceleration of the floor, sliding on PTFE plates, was monitored.

10.5.1 Oblique foot impacts Experiments were performed on a 40 mm thick PU225 mat and on a 50 mm thick mat of crosslinked polyethylene foam of density 24 kg m⫺3 (PE24). Table 10.5 shows that the peak impact vertical force was


Peak forces and rotations, for vertical drop of 0.6 m

Mat

Thickness (mm)

Peak FV (kN)

Peak FH (kN)

Rotation deg at peak FV

40 50

3080 1620

580 250

6 17

PU225 PE24

4

0 Rotation angle

2

20 Vertical force

1

0 830

40

Horizontal force 835

840

845 Time (ms)

Rotation angle (°)

Force (kN)

3

850

855

860

Figure 10.11 Variation of normal and shear force, and ankle joint rotation with time (Mills and Lyn, 2004).

lower with the PE24 mat, a result of the longer duration impact. As the horizontal FH and vertical FV forces have different time dependences (Fig. 10.11), it is not possible to measure a friction coefficient. The foot bounces from both mats.

10.5.2 FEA of mat deformation The hyperfoam material model in ABAQUS uses the Ogden strain energy function; a good fit to both the compressive and shear impact data (see Fig. 5.14) for PU225 was obtained with shear moduli µ1 ⫽ 225 kPa, µ2 ⫽ 7 kPa, strain energy function exponents α1 ⫽ 7, α2 ⫽ ⫺2, and Poisson’s ratios ν1 ⫽ 0.22, ν2 ⫽ 0.45. Mats made from PU98 and PU225 foam were modelled. A twodimensional problem was tackled using plane strain elements to reduce the solution time. This simplification of foot geometry ignores effects at the toe and heel, so probably slightly underestimates the forces. The foot was taken to be 100 mm wide with a 57 mm wide flat region on the plantar surface, and a radius of curvature at the edges


249

20 40 60 100

80

20

40

60

60

40

20

Figure 10.12 Predicted shape and compressive principal stress contours (kPa) when a 60 mm thick PU98 mat is compressed by a foot vertical load of 2 kN and a moment of 20 Nm (Mills and Lyn, 2004).

Table 10.6

FEA predictions of mat deformation for 2 kN, 20 Nm, and friction 0.4 Foam

Mat thickness (mm)

Indentation (mm)

Twist angle (°)

PU225 PU225 PU98

40 60 40

5.5 7.3 25

6 7 20

of 30 mm. Its Young’s modulus was taken as 5 MPa to allow a little deformation. The mat was first compressed by a typical load for a foot strike; a vertical loading of 2 kN on a plane strain model of thickness 0.3 m. In the simulation, the mat resistance to foot eversion (rotation about a toe–heel axis) was evaluated when a moment of 20 Nm was applied to the heel–toe axis of the foot. Figure 10.12 shows that a 20 Nm moment is predicted to cause considerable foot supination as the foam deforms. The predicted rotation angle is 3 times larger for a 60 mm mat of PU98 (Table 10.6). The foot oblique landing tests show that judo mat foams differ in their response, with higher impact forces for the dense PU foam mat, than for the closed-cell PE foam.

References BS EN 960 (1995) Headforms for use in the testing of protective helmets, British Standards Institution, London. BS EN 12503 (2001) Parts 1 to 7, Sports Mats, British Standards Institution, London. Hulme A.J. & Mills N.J. (1996) The Performance under Impact of Industrial Helmets, Research report no. 91, Health and Safety Executive, Sheffield.

250 Polymer Foams Handbook Lyn G. & Mills N.J. (2001) Design of foam crash mats for head impact protection, Sport. Eng. 4, 153–163. Mills N.J. (2006) Finite element models for the viscoelasticity of open-cell polyurethane foam, Cell. Polym. 25, 277–300. Mills N.J. & Lyn G. (2001) Design of foam padding for rugby posts, in Materials in Science and Sports, Ed. Froes F.H., TMS, Warrendale, PA, pp. 105–117. Mills N.J. & Lyn G. (2004) Design and performance of Judo mats, in The Engineering of Sport 5, Vol. 2, Eds. Hubbard M., Mehta R.D. & Pallis J.M., ISEA, Sheffield, pp. 495–502. Wright I.C., Neptune R.R. et al. (2000) The effects of ankle compliance and flexibility on ankle sprains. Med. Sci. Sport. Exerc. 32, 260–265.

Chapter 11

Micromechanics of closed-cell foams

Chapter contents 11.1 11.2 11.3 11.4 11.5 11.6

Introduction Observations of cell deformation Material responses Air–polymer interactions Kelvin foam model for elastic moduli Regular foam models for high strain compression 11.7 Irregular foam models 11.8 Bead foams 11.9 Discussion References

252 253 254 256 262 267 274 275 276 277


11.1

Introduction Closed-cell foams made from polystyrene (PS), polyethylene (PE), and polypropylene (PP) are mainly used for packaging or impact protection. Their mechanical properties are dominated by the polymer contribution. In contrast, the mechanical properties of soft closed-cell foams, such as ethylene–vinyl acetate (EVA) copolymer foams used in the midsoles of training shoes, are dominated by the contribution of the cell gas because the polymer has a low crystallinity, and consequently a low Young’s modulus. The prime subject of modelling is the foam compressive response. Building on the theory given in Chapter 7, further mechanics concepts are introduced (Fig. 11.1). After a review of deformation mechanisms, elastic moduli are considered. These are relevant to the foam cores of sandwich beams, where elastic deflection is a design criterion. Moduli are considered before the modelling of high strain compressive response. The mechanics of closed-cell thermoplastic foams involves three interacting levels: 1. Polymer microstructure which affects its mechanical response. 2. Cell edges which can be considered as beams and faces which can be considered as membranes. 3. Cell geometry is the statistics of shape and size, and the measures of regularity.

Figure 11.1

Feature of foam

Theory

Cell faces

Parallel load paths

Air

Isothermal gas compression in slow tests

Air–polymer interactions

Face bowing Air diffusion through faces Heat transfer air to face Face crumpling if the cell shrinks

Polyhedral cells

Elastic moduli of dry Kelvin foam Model of supported edges in Kelvin foam FEA of dry Kelvin foam [001] compression

Irregular cell shapes

Voronoi models

Bead boundaries

FEA of bead deformation

Concepts used to explain the micromechanics of closed-cell foams.

Chapter 11 Micromechanics of closed-cell foams

11.2

253

Observations of cell deformation It is more difficult to observe the deformation of closed-cell than opencell foams. The cell faces scatter light, preventing optical microscopes from viewing the interior. The cell faces are usually too thin to be resolved by X-ray microtomography. The process of cutting through the foam severely weakens the cut cells, so deformation mechanisms in external part cells may be atypical (Fig. 11.2). The response varies with the polymer. Permanent deformation mechanisms can be identified if the deformed foam is sectioned without causing further damage, and compared with the original state. High-density PE (HDPE) foams of density 50 kg m3, when compressed to 80% strain, suffered permanent strains of 10% (Loveridge and Mills, 1991). There were wrinkled faces in the foam after recovery, due either to permanent stretching of the faces or permanent bending of some of the thicker edges. These must be distinguished from wrinkled cell faces that result from processing; Figure 3.12 showed wrinkled faces in lowdensity PE (LDPE) foams after 24 h under vacuum (Masso-Moreu and Mills, 2004). Figure 1.5 showed that polyolefin foams of density circa 90 kg m3 can have thick ‘super-vertices’ between some cells, which may affect the deformation pattern. Plastic hinges occur in PS foam cell faces if the compressive strain exceeds 10% (Mills, 1994). The permanent strain is approximately half of the maximum foam compressive strain, for PS bead foam of density 50 kg m3. As foam processes have improved, the cell faces tend to have more uniform thickness and there is less polymer in the vertices. This improves the recovery after large compressive strains.

100 µm

Figure 11.2

SEM photograph of cut cells on the surface of EPS, compressed in the direction of the arrows (Mills, unpublished).


11.3 Material responses 11.3.1 The gas and polymer in parallel Skochdopole and Ruben’s (1965) qualitative model (Fig. 11.3) suggests the cell air and the polymer structure acting in parallel when the foam is compressed. Hence the gas and the polymer structure undergo the same strain. The model adds the stress σP borne by the polymer structure to the stress σG borne by the cell gas. It assumes that there is no synergistic effect of increasing gas pressure on the strength of the polymer structure. Although they did not express the contributions as functions of strain, their model is the basis of most subsequent models. The foam dimensions are usually more than an order of magnitude larger than the cell dimensions. Consequently, cut surface cells or those containing a pressure lower than in the interior (see Section 4.1) can be ignored when considering the foam properties. Hence the mean cell diameter has little effect on the polymer term.

11.3.2 Polymer response Modelling predicts that high tensile strains occur in cell faces, but it is not currently possible to obtain tensile stress–strain data from an individual cell face. Chapter 3 detailed how a Young’s modulus of 202 MPa was measured for biaxially oriented LDPE packaging film at a low strain rate. With allowance for the increase in modulus with increasing strain rate, a Young’s modulus of 300 MPa was used for modelling the deformation of LDPE faces at high strain rates. For foam modelling, the polymer is assumed either to be linearly elastic and isotropic with Young’s modulus E and Poisson’s ratio ν or to be elastic–plastic with a yield stress that increases with strain.

Force

Gas

Figure 11.3

Cell walls

Model (redrawn from Skochdopole and Rubens, 1965) of the cell air and the polymer structure acting in parallel in a compressed closed-cell foam.

255


11.3.3 Air response Rusch (1970) suggested that the contribution σG of the cell gas pressure to the compressive stress, when a closed-cell foam is uniaxially compressed, is given by the expression σG

paε f 1 ε R

(11.1)

where pa is atmospheric pressure (the assumed gas pressure in the undeformed foam cells), ε is the applied compressive strain, f is the fraction of closed cells in the foam, and R is the foam relative density. Although not stated, his assumptions were zero lateral expansion of the foam and isothermal gas compression. Figure 11.4 shows the polymer and gas in 1 m3 of foam, separated so that the relative volumes can be seen. For foam with zero Poisson’s ratio, the compressive strain ε is also the volumetric strain. The polymer is taken to be incompressible, so the volume of gas in the compressed foam is 1 ε R. The absolute pressure p of the isothermal gas in the compressed foam is given by pa (1 R) p(1 ε R)

(11.2)

The compressive stress σ is measured relative to atmospheric pressure pa so σ p pa pa

Gas at pressure pa 1−R

ε (1 ε R)

Stress σ

Gas at pressure p Polymer

R

Polymer

(11.3)

ε

1−ε−R R

σ

Figure 11.4

Phase volumes in a foam of zero Poisson’s ratio and relative density R, before and after uniaxial compressive strain ε.

256 Polymer Foams Handbook Gibson and Ashby (1988) used equation (11.1) with f 1, and assumed that the polymer contribution σ0 was constant, obtained σ σ0

paε 1 ε R

(11.4)

The expression ε/(1 ε R) will be referred to as the gas volumetric strain. σ0 can be evaluated by fitting a graph of stress against the gas volumetric strain, and extrapolating to zero strain. Clutton and Rice (1991) fitted the compressive stress–strain curves of polyolefin foams with this equation, but found that the value of pa often did not equal 101.3 kPa (standard atmospheric pressure). Hence, it is often written p0 and called the effective gas pressure in the cells. If the foam has a constant Poisson’s ratio ν, the compressive stress is given by ⎛ ⎞ 1 R 0 pa ⎜⎜ 1⎟⎟⎟ ⎟⎠ ⎜⎝ (1 ε)(1 νε)2 R

11.4

(11.5)

Air–polymer interactions 11.4.1 Face bulging due to cell pressure differentials When a foam block is under a compressive load, as a result of gas diffusion, the gas pressure in boundary cells will be lower than in the interior. Unless the foam is bonded to a metal surface, there is a sufficient gap at metal/foam boundaries for gas to escape from cut cells. Consequently surface cells are more highly deformed than interior cells. A pressure differential ∆p between two cells causes the intervening face to be curved. Assuming that a face of thickness δ becomes a spherical cap with radius of curvature r, it contains a biaxial tensile stress

1 2

∆pr 2δ

(11.6)

Since polymers have a finite permeability P to gas, there will be a gas flow rate Q m3 s1 across a face of area A Q

PA ∆p δ

(11.7)

During typical compressive loading, lasting a few minutes, the gas escape from the sides of a block of side 20 mm is insufficient to significantly alter the response. There will be a steep pressure gradient near the surface of the foam block, causing tensile stresses in some cell faces.


257

In the interior of the block there will be a uniform pressure, and flat cell faces. However, in creep tests lasting many hours, gas loss from PE and PP foams must be considered (Chapter 19).

11.4.2 Foam diffusivity The diffusivity of closed-cell foams to gases or liquids can be related to the foam structure and the polymer diffusivity. Gas diffusion affects creep, where the load is a applied for long times (Chapter 19), and thermal conductivity, since the blowing agent may have a lower conductivity than air (Chapter 15). Water diffusion into foam (Chapter 20) may also affect mechanical properties. Ostrogorsky et al. (1986) considered the relationship between the gas diffusion coefficient of the polymer Dp, the cell geometry, and the diffusion coefficient Df of a foam. They calculated the foam permeability, and multiplied this by the solubility S of the gas in the cells (m3 at STP m3 atm1) to obtain the foam diffusion coefficient as DF 2

L D S δ P

(11.8)

– where L is the mean distance between faces in the direction of diffusion and δ is the face thickness. The polymer of a typical low-density foam dissolves a negligible amount of gas, so S 1. The factor 2 arises because the average projected area of a cell face, onto the plane normal to the pressure gradient, is half the cell face area. The physics behind this equation is demonstrated in an analysis using a simple cubic-cell model (Mills and Gilchrist, 1997a). Figure 11.5 shows three neighbouring cube-shaped cells containing gas at pressures pi1, pi, and pi1 with a pressure gradient acting along the x-axis. Substituting a typical cell face thickness δ 1 µm and oxygen diffusion constant for LDPE Dp 0.5 1010 m2 s1

L δ Pi −1

Pi Qin

Pi +1 Qout

x

Figure 11.5

The diffusive gas flows between three neighbouring cubic cells.

258 Polymer Foams Handbook in the equation for the time lag before steady state flow develops (Mills, 2005) tL

δ2 6DP

(11.9)

gives a 3 ms time lag. Hence, steady state gas flow occurs for loading times 0.01 s. The gas flow rate Q (m3 at STP s1) through the face between cells i and i 1 is Q PA

pi pi 1 δ

(11.10)

where P is the polymer permeability and A is the face area. Using versions of equation (11.10) for the faces to the left and right of the central cell, and replacing A by L2, gives the net transfer of gas ∆Q into the central cell in a time interval ∆t ∆Q PL2 (pi 1 2pi pi 1) ∆t δ

(11.11)

The pressure increase ∆p in the central cell is related to the net gas flow by ∆p

pa∆Q L3

(11.12)

where pa is atmospheric pressure. Hence the pressure change in cell i is given by ∆p

Pp ∆t (pi 1 2pi pi 1) δL

(11.13)

This is the basis of an explicit finite difference equation for the diffusion process. The dimensionless gas diffusion constant D is defined by D

Ppa∆t Ppa∆t δL δ∆x

(11.14)

If a solid with a diffusion coefficient Df is divided into layers of thickness ∆x for finite difference computation, D is related to the time interval by D Df

∆t ∆x2

(11.15)


259

Comparing equations (11.14) and (11.15) shows that ⎛L⎞ Df Ppa ⎜⎜ ⎟⎟⎟ ⎜⎝ δ ⎟⎠

(11.16)

This is the same as equation (11.8) without the factor 2. To re-express equation (11.16) in terms of the foam relative density R, the fraction of polymer φ in the cell faces must be known. For a cubic-cell foam, the relative face density is φR

3L2δ 3δ 3 L L

(11.17)

while for a Kelvin foam of cell diameter L, the constant in the equation is 3.35. Expressing equation (11.8) in terms of R, the foam diffusivity is Df

6Ppa φR

(11.18)

11.4.3 Heat transfer from cell air to faces Throne and Progelhof (1984) stated without justification that foam compression is adiabatic, whereas Gibson and Ashby (1988) stated without justification that it is isothermal. Burgess (1988) considered the effect of both assumptions on the shape of cushion curves, and concluded that the conditions are not adiabatic. He was unable to solve the heat transfer equations for intermediate cases. The physics of heat transfer in small cells, which deform on a 1 ms time scale, has not been investigated experimentally. Analysis of foam thermal conductivity suggests that convection does not occur in closed cells of diameter smaller than 10 mm. Mills and Gilchrist (1997b) modelled heat transfer, allowing for cell dimension changes as the foam is compressed. In this coupled problem, cell deformation affects the time scale of heat transfer and the gas heating affects its pressure. The cells are approximated as cubes, compressed along the x-axis. Once the compressive strain exceeds 30%, the majority of the heat conduction will be along the x-axis. Therefore, a one-dimensional finite difference heat transfer analysis was made, with mirror symmetry planes, across the cell centre and through the face midthickness, the face thickness δ. The thermal diffusivity D of air is related to its thermal conductivity k, specific heat cp, and density ρ by D

k ρcp

(11.19)

260 Polymer Foams Handbook D 1.88 mm2 s1 at STP. When air is compressed, the thermal conductivity should increase linearly with density, and the specific heat (J kg1 K1) should stay constant, so D is assumed to remain constant. Polymers have much lower thermal diffusivities, typically 0.1 mm2 s1, but the cell faces are very thin. The heat flow problem is for two slabs of material, one of varying thickness, connected at the air/polymer interface. The Fourier number Fo (the dimensionless time) is, for a slab of thickness 2L heated from both sides for a time t Fo

Dt L2

(11.20)

Since the cell face half thickness is typically 3 µm for a foam of cell diameter 200 µm and relative density 0.05, the cell face Fourier number is 44 after 1 ms, compared with 0.19 for the air uncompressed cell, or 4.7 in a cell compressed by 80%. Therefore, the polymer remains at a uniform temperature, with the temperature gradient in the cell air. A drop test of the type used for determining cushion curves (Chapter 12) was modelled: a body of mass m 5 kg, protected by a 50 mm thickness of foam of area A 25,000 mm2, fell with an impact velocity V0 7.0 m s1 onto a flat rigid surface. The static stress on the foam was 2 kPa. The foam was assumed to have a relative density of 0.05, zero Poisson’s ratio, and a negligible initial yield stress σ0. Its initial temperature was T0 and the initial gas pressure p0 was atmospheric. The time interval ∆t between heat conduction calculations is chosen to be the smallest that allows stable computations. The steps, repeated at time interval ∆t1 0.1 ms, were to calculate: 1. The new height of the block (hence the strain ε) from the striker velocity. 2. The adiabatic increase in the absolute temperature of the cell air Tnew ,i Told,i

⎛ V ⎞⎟ 1 ⎜⎜⎜ old ⎟⎟ ⎝⎜ Vnew ⎟⎠

(11.21)

where the suffixold refers to the values at the previous time step. The average air temperature before the diffusion calculation, – TB, is stored. 3. Thermal diffusion using the explicit finite difference formula for layers 2 to n Tnew (i) 0.5(Told (i 1) Told (i 1))

(11.22)

where the new temperature is that after a time interval ∆t. A special recurrence relationship is used for cell layer 1. The temperature


261

of the zero layer, representing the polymer, is unaffected, and the temperature of the n 1th cell layer, across the symmetry plane through the gas, is equal to Tnew(n). The process is repeated until the elapsed time has reached ∆t1, then the average gas temperature – T A is calculated. 4. The heat gained by the polymer is a numerical integral of the change in gas heat content. The new polymer temperature, assuming the heat is shared among two cell faces, is P (0) T P (0) Tnew old

LρA cpA δρP cpP

(TB TA )

(11.23)

5. The air pressure p from the average temperature and the overall volume p p0

TA V0g T0 Vg

(11.24)

6. The acceleration a of the falling object from the foam compressive stress a

(p p0 )A m

(11.25)

7. The new velocity V by numerical integration of the acceleration.

15

150

5

Polymer

0

500

−5 −10

Stress

0

5

(a)

Figure 11.6

10 15 20 Time (ms)

250 25

0 30

(b)

100 Cell air

50 0 −50

Stress

200

Polymer

0

5

10 15 Time (ms)

400

20

Stress (kPa)

10

Temperature (°C)

Cell air

Stress( kPa)

Temperature rise (°C)

The predicted maximum strains are about 80%, and the maximum acceleration of the packaged object is about 250g. Figure 11.6

0 25

Predicted temperature vs. time graph for impact (a) 0.1 mm cells and (b) 1.0 mm diameter cells (Mills and Gilchrist, 1997b).

262 Polymer Foams Handbook compares the time variation of the air and polymer temperatures during the stress cycle. When the cell diameter L is 0.2 mm (Fig. 11.6a), the air temperature rises by 11°C early in the compression cycle, when heat diffusion distances are still large. The polymer heats by 9°C, in phase with the stress cycle, and the conditions are almost isothermal. When the cell diameter is 1.0 mm (Fig. 11.6b), the air temperature rises by 93°C, but returns to ambient just after the peak stress, since heat diffusion is rapid in the highly compressed foam. When the foam expands, the air cools 40°C below room temperature, but the polymer temperature hardly changes. Finally, for cells of 4 mm diameter, the heat transfer is so poor that the air heats nearly adiabatically by a maximum of 189°C, and the temperature cycle is once again in phase with the stress cycle. The adiabatic heating limit, calculated using the equation pT 1 constant

(11.26)

is an air temperature rise of 210°C for a strain of 70.5% in a foam of relative density 0.05. The predicted stress vs. gas volumetric strain relationship, for 0.2 mm diameter cells, has hardly any hysteresis (Fig. 11.7a). The slope of the plot, representing the effective cell pressure p0, is 74 kPa for effectively isothermal conditions. When the cell diameter is 1 mm, there is significant hysteresis in the stress–strain curve, due to the heat flows described. For large cells of 5 mm diameter, the predicted stress–strain curve again has little hysteresis. For commercial foams, conditions are nearly isothermal at impact strain rates. Impact test data for LDPE foam of density 23 kg m3, and mean cell diameter 1.32 mm is shown in Figure 11.7b. The effective gas pressure was 70 kPa, showing that there are negligible effects of cell gas heating. The difference between the loading and unloading response is due to the polymer contribution being non-reversible, so the best-fit lines for loading and unloading have positive and negative intercepts on the stress axis.

11.5

Kelvin foam model for elastic moduli 11.5.1 Young’s modulus Renz and Ehrenstein (1982) used finite element analysis (FEA) of the representative unit cell (RUC) shown in Figure 11.8 to predict the Young’s modulus of a closed-cell Kelvin foam in the [001] direction. There is mirror symmetry at all the boundary planes, and the polymer is entirely in the cell faces. Their graphical results, for the variation of the


263

0.5

Stress (MPa)

0.4 0.3 0.2 0.1 0

0

1

2 3 Gas volumetric strain

4

5

0

1

2 3 Gas volumetric strain

4

5

(a) 0.4

Stress (MPa)

0.3

0.2

0.1

0 (b)

Figure 11.7

Impact stress–strain curves: (a) predicted for foam cell diameters of 0.2 mm (dashed) and 1.0 mm (solid curve) and (b) experimental for LDPE foam of density 24 kg m3 and average cell diameter 1.32 mm; the loading data is fitted with a straight line (Mills and Gilchrist, 1997b).

Young’s modulus Ef of polyvinylchloride (PVC) foam with relative density R, follows a linear relationship Ef 0.33ER

(11.27)

where E is the polymer Young’s modulus. This result was confirmed by Kraynik et al. (1999), who found the constant in equation (11.27) to be 0.299 for compression in the [001] direction, and 0.322 for compression in the [111] lattice direction. Figure 11.10 shows the predicted Young’s modulus is considerably higher than data for PS foams.

264 Polymer Foams Handbook z

F

F

Quarter of square face

y

Edge x

F

F

Figure 11.8

External forces on the RUC of a closed-cell Kelvin foam extended in the [001] (z) direction (redrawn from Renz and Ehrenstein, 1982, with a finite edge width).

Therefore, either face buckling occurs at very low foam strains, or cell faces are wrinkled as a result of gas diffusion after processing.

11.5.2 Bulk modulus Almanza et al. (2004) calculated the bulk modulus of the Kelvin foam with flat faces. Equation (3.2) gives the biaxial tensile stress σf in hexagonal faces, as a function of the relative pressure in the Kelvin foam cells. The polymer is assumed to be linearly elastic, so the tensile elastic strain in each face εE is εE (1 ν)

σf E

(11.28)

where E is the polymer Young’s modulus for stresses acting in the plane of the face and ν its Poisson’s ratio. If equation (3.2) is divided by the tensile strain in the face (which is also the tensile strain in the foam), and equation (11.28) used to replace σf /εE, this gives 3(pC P) ER 1ν 2εE

(11.29)

where pC is the cell air relative pressure and P is the hydrostatic pressure applied to the foam.


265

For small volume strains, the foam volume strain εV 3εE, and the foam bulk modulus KF is defined by KF ≡

P εV

(11.30)

The bulk modulus KA of the air in the cells is KA ≡

pC p0C εV

(11.31)

where p0C is the absolute air pressure in the cells. For foams stored at atmospheric pressure for several months, p0C pa the atmospheric pressure. Consequently the foam bulk modulus KF KP KA

2ER pa 9(1 ν)

(11.32)

where KP is the contribution of the polymer structure. Masso-Moreu and Mills (2004) found that LDPE foam of density 19 kg m3, with slightly wrinkled cell faces, had a bulk modulus of 440 kPa for relative pressures in the range 0 to 20 kPa, but 820 kPa for relative pressures in the range 40 to 80 kPa. Substituting E 202 MPa and ν 0.4 in equation (11.32), the theoretical value of KF is 1550 kPa. Hence the experimental bulk modulus is lower than the theoretical value, confirming the effect of wrinkled cell faces.

11.5.3 Young’s modulus assuming wrinkled faces Mills and Zhu (1999) modelled the compressive response of the Kelvin model. They assumed that the faces have no resistance to wrinkling, if their width decreases under load. The foam contains sufficient edge material (φf 0.6) for it to act as the main structure, supported by faces in tension. As this is marginally realistic for most PE and PS foams, the model will only be described briefly. It however provides a lower bound for the foam Young’s modulus. The faces are considered as membranes, which cannot transmit compressive stresses to the edges. Consider a face of uniform width w and thickness δ, with built-in ends a distance L apart. The compressive force Fc to initiate elastic buckling is given by ⎛ 2π ⎞2 wδ 3 Fc ⎜⎜ ⎟⎟⎟ EI where I ⎜⎝ L ⎟⎠ 12

(11.33)

266 Polymer Foams Handbook F z

F B

√2L

fs Square face

fH Hexagonal face

C

y 2L

Figure 11.9

2L

Structure cell with mirror plane boundaries.The horizontal tensile forces in faces connected to edge BC are shown (redrawn from Mills and Zhu, 1999).

The critical strain to cause buckling is therefore 2

εc

Fc 1 ⎛ π ⎞ ⎜⎜ ⎟⎟⎟ Ew

3 ⎜⎝ L ⎟⎠

(11.34)

For a relative density of 0.025, and face fraction 0.9, δ/L 0.0189 and the strain to cause buckling is 0.11%. Consequently, the faces will be wrinkled at the typical 1% strain used to measure Young’s modulus. Figure 11.9 shows the structural cell for the Kelvin foam compressed in the [001] lattice direction, which is bounded by mirror planes. The vertical square face is considered to consist of horizontal ribbons, which exert a force fS on the edge BC. The hexagonal face acts as a set of horizontal ribbons which exert a horizontal force fH. Two hexagonal faces exert tensile forces of the same magnitude on BC, and their vector sum f2 acts along the negative y-axis. Consequently, the edge BC bends under the influences of the vertical force F, and face tensions fS and f2 which vary along its length. The edge shape was found by iterative methods, and cell pressure changes were also considered. For a LDPE foam relative density of 0.025, when the relative face density φ was increased from 0.2 to 0.65, the predicted Young’s modulus stayed in the range 400–460 kPa. The Young’s modulus, predicted for PS foam using φ 0.6, is given as a function of the foam relative density R by Ef 0.0598ER1.066

(11.35)

Foam Young’s modulus (MPa)


267

Kraynik

1

Experiment Mills and Zhu 0.1 0.01

0.1 Relative density

Figure 11.10 Predicted variation of Young’s modulus of PS foam with density, compared with experimental data for EPS foam and FEA predictions (Mills and Zhu, 1999).

It is compared in Figure 11.10 with experimental Young’s moduli, which vary with relative density according to Ef 0.977 ER1.627

(11.36)

The Young’s modulus of PS E 3.0 GPa. Figure 11.10 also shows the FEA prediction of equation (11.27). Neither model fits the experimental data, but they act as upper and lower bounds. The Mills and Zhu model predictions are good at low relative densities, where the face buckling strain is low, while the data approaches the FEA predictions for flat faces at R 0.1. Simone and Gibson (1998) predicted the effect of wrinkled cell faces (in aluminium closed-cell foams), on the Young’s modulus, by FEA of a modified Kelvin foam. This showed that the Young’s modulus could be reduced by a factor of 10 from the flat-face version. Face wrinkling, prior to loading, is also likely to be a significant for LDPE foams. The gas contribution dominates the Young’s modulus for closed-cell foams from flexible polymers with relative densities 0.1, while it is insignificant for higher-density foams made from high modulus polymers.

11.6

Regular foam models for high strain compression 11.6.1 Introduction At present, micromechanics models for the compression of closedcell rigid foams are incomplete. The cell faces, which usually

268 Polymer Foams Handbook contain 80% of the polymer (Chapter 3), provide the majority of the polymer contribution to the stress. Two factors affect the face deformation mechanisms: (a) The cell gas pressure. If the volumes of neighbouring cells decrease by different amounts, the resulting pressure differential causes the intervening face to bow. However, if cell faces concertina as in a bellows, and the number of half waves in the buckled shape is even, the volume of each cell is equally compressed. (b) Cell faces are stabilised by the attached faces, and the deformation of these faces is related. However, tensile plastic deformation near the linking edge can decouple the deformation modes, as explained in Section 11.6.4.

11.6.2 Gibson–Ashby models Figure 11.11a shows the model for high strain deformation of closedcell elastic foams used by Gibson and Ashby (1988). As discussed in Chapter 7, the model geometry differs from that of polymer foams. The added faces create enclosed cubic cells, but leave surrounding open regions. When the model is used for the foam Young’s modulus, they argue that the cell faces contribute a term C2φR, where C2 is an unknown constant and φ is the fraction of polymer in the cell faces. They assume incorrectly that the relationship is valid for 1 φ 0, so F

F

F Face yield in bending

Face

Face tensile yield

Edge

F F

F (a)

F

(b)

Figure 11.11 Gibson and Ashby’s models for (a) elastic deformation of closed-cell foam and (b) plastic deformation of a honeycomb with end faces (redrawn).


269

as Ef EP when φ 1, they deduce that C2 艑 1. Hence, fitting the relationship to Young’s modulus data for foams, they deduce that φ lies in the range 0.2–0.4. However, using Renz and Ehrenstein’s (1982) prediction for small φ (equation (11.27)), which is equivalent to C2 0.33, the (scattered) experimental data is consistent with the experimental φ 0.7 for low-density polyolefin foams. When they use the model to calculate the collapse stress σ*el of elastic foams, they ignore the contribution of the faces, and assert (as explained in Chapter 7) that the polymer structure collapses at a constant stress when axially compressed cell edges buckle. Hence, with the cell gas contribution (equation (11.3)), the collapse stress is * C ER2 σel 4 e

p0ε 1 ε R

(11.37)

where C4 is an empirical constant and Re is the volume fraction of the cell edges. By fitting this equation to initial yield stress data for PE and PS foams, they deduced that C4 艑 0.05. However, the non-linear effects on unloading PE foams (Fig. 11.7b) cannot be explained by an elastic model. Hence their generalisation that LDPE foams are elastomeric, collapsing elastically at a stress 艑0.05ER2, is not justified. The model (Fig. 11.11b) for foams that plastically deform is the inplane loading of a honeycomb with closed end faces, not a 3D foam structure. It emphasises the plastic hinges at the ends of edges, and tensile yield in faces in a direction perpendicular to the foam compressive stress, but ignores elastic edge bending. Hence, it is a pictogram of deformation mechanisms rather than a micromechanics model for closed-cell foams. It was used to derive a scaling relationship for the initial collapse stress σ*pl C5YRe1.5 C6YRf

(11.38)

where C5 and C6 are empirical constants, Y is the polymer tensile yield stress, Re and Rf are the volume fractions of polymer in the cell edges and faces. A fit of this equation, to scattered yield stress data for PS, PVC, and rigid polyurethane (PU) foams, lead to the tentative conclusion that C5 0.3 and C6 0, implying that the cell faces have no effect on the collapse stress. They argue that the fraction φ of material in the faces may be near zero, or that cell faces may rupture before the foam fully collapses (neither of which are realistic). Although the predicted scaling relationship with Re is reasonable, the model geometry and mechanisms are inappropriate for closed-cell thermoplastic foams. Unfortunately, some researchers have placed more reliance on this theory than on experimental evidence, so have used φ as a curve-fitting


Stress (kPa)

200

150 Impact data

100

Prediction

50 Polymer contribution 0

0

10

20

30

40 50 Strain (%)

60

70

80

Figure 11.12 Predicted compressive stress–strain curve and polymer contribution, for LDPE

foam of R 0.025, E 300 MPa, face fraction 0.6, face yield stresses Y1 12 and Y2 10 MPa, and elastic edges, compared with impact loading data for an LDPE foam of the same relative density (Mills and Zhu, 1999).

parameter and deduce that the foam studied has a low φ value. Chen et al. (1994) concluded that polymethacrylimide foam had φ in the range 0.3–0.4, when scanning electron microscope (SEM) photos of the microstructure (Li et al., 2000) show that nearly all the polymer is in the faces (see also Table 11.1, where a value of 0.6 was measured).

11.6.3 Kelvin foam elastic–plastic model LDPE foam of relative density 0.025 was modelled (Mills and Zhu, 1999) using E 300 MPa and φ 0.6. The tensile response of the polymer was assumed to be If ε εy σ Eε If ε εy σ Y1 εY2

(11.39)

where the yield stresses Y1 12 MPa and Y2 10 MPa. The edge BC (Fig. 11.9) bends as the foam strain increases, causing the structure cell to expand laterally. The Poisson’s ratio at high foam strains is predicted to be 0.2. Figure 11.12 compares the predicted stress–strain curve with experimental data for LDPE foam of density 24 kg m3. The agreement of the loading curve with the theory is excellent, but cell air compression provides the majority of the stress. A


271

Compression axis t Cruciform section Pyramidal section

h

Extension Plastic zone hinges

b

Figure 11.13

Hinge formation in the cruciform section of a ‘truncated cube’ model of rigid foam compressive yield (adapted from Santoza and Wierzibicki, 1998). y P Stress

Figure 11.14 A plastic hinge in a beam and the stress distribution (solid lines) in the hinge compared with that in a non-linear viscoelastic material (dotted curve).

similar model for PS foam overestimated the collapse stress by a factor of 2.

11.6.4 Wierzibicki plasticity model The plastic hinge pattern in compressed PS foam (Fig. 11.2) is similar to one predicted in Santoza and Wierzibicki’s (1998) model for the crushing of aluminium closed-cell foam. The RUC of the lattice of truncated cubes consists of a cruciform section and a pyramidal section (Fig. 11.13). The former is assumed to form plastic hinges as shown, when the model is compressed along a cube axis. The required plastic bending moments Mpl for a face w wide and d thick (Fig. 11.14) are related to the initial yield stress σ0 of a material that does not harden by Mpl

wd 2 0 4

(11.40)

272 Polymer Foams Handbook The rate of plastic energy dissipation is minimised to obtain the plastic hinge spacing. Consequently, the mean compressive stress to crush the foam is ⎛ t ⎞1.5 f 4.43 0 ⎜⎜ ⎟⎟⎟ ⎜⎝ b ⎟⎠

(11.41)

where t is the face thickness and b its breadth. If further terms for the crushing of the pyramidal section are ignored, the result, expressed in terms of the foam relative density R, becomes

f 0.63R1.5 0

(11.42)

Experimental plots of compressive yield stress vs. density for expanded polystyrene (EPS) (Mills, 1994) and HDPE foams (Clutton and Rice, 1991) on logarithmic scales have slopes 艑1.5. If the lines are extrapolated to the density of the solid polymer, the yield stress is close to the σ0 measured for the polymer at low strain rates. Consequently the predictions of equation (11.42) are confirmed. Bureau et al. (2005) show that the cell size of LDPE foams has some effect on the initial yield stress, possibly because of variations in the amount of orientation in the cell faces. However, there did not appear to be a cell size effect on the initial yield stress of extruded PS foams (Bureau and Gendron, 2003). For HDPE cell faces, which behave in a non-linear viscoelastic manner, the stress distribution in a bent cell face (Fig. 11.14) is similar to that in a plastic hinge, so it is not surprising that the exponent in the yield stress–relative density relationship is the same as for PS.

11.6.5 Dynamic FEA of Kelvin foam model McKown (2005) examined a model with two layers of Kelvin cells in the [001] compression direction, with mirror symmetry in the lateral directions (Fig. 11.15). The gas pressure in the cells was ignored, since the model was initially for aluminium foam. Dynamic FEA predicted that the top cell layer, hit by the striker, collapsed by face crumpling before the lower layer. The predicted compressive stress was almost constant with strain, but it was markedly higher than that observed in PS foams. Therefore it is likely that either cell shape irregularity or cell face wrinkling, is responsible for the lower experimental collapse stress. Nevertheless, this model shows possible deformation mechanisms in a foam with 100% of the polymer in the faces. For closed-cell


273

Time = 0.045 Time = 0.09

Time = 0.135

Time = 0.16

Stress (MPa)

(a) 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0.0 (b)

0.1

0.2

0.3

0.4 0.5 Strain

0.6

0.7

0.8

0.9

Figure 11.15 (a) Predicted face crumpling with plastic hinges in EPS at the times indicated and (b) stress–strain curve (McKown, 2005).

polymer foams, consideration of the cell pressures would modify the predicted deformation pattern.

11.6.6 Modelling unloading Impacted PS foams have permanent strains, which increase with the maximum compressive strain (Fig. 11.16). Plastic hinges form in the faces, and the rotation of these hinges increases as the compressive strain increases. The reverse deformation process is driven by the gas pressure in the cells, and resisted by the plastically deformed faces and edges. If the compressive strain is 50%, the cell gas pressure is


Residual strain (%)

40 35 30 25 20 15 10 20

30

40

50

60

70

80

90

Maximum strain in impact (%)

Figure 11.16 Permanent strain vs. maximum impact strain for PS foam of density 65 kg m3 (Mills, 1996).

100 kPa above atmospheric. However the stress to cause yielding, in PS foams of density exceeding 30 kg m3, is greater, so there is no reverse yielding of the plastic hinges. At present modelling has not considered unloading.

11.7

Irregular foam models Shulmeister (1998) used FEA to predict the Young’s moduli of face and body centred cubic (FCC and BCC, respectively) lattices of cells into which various amounts of disorder were introduced. The faces were assumed to be flat. The cell nuclei at the boundaries of their RUC, of edge LC, were regularly spaced, but those at a depth greater than d into the cube were allocated random positions. The proportion of random cells in the cube (1 d/LC)3 was increased from 0% to 58% and the trends in the predicted moduli examined. For the modified BCC foams, with a fraction φ 0.9 of material in the faces and relative density R 0.025, the randomness only decreased the Young’s modulus by about 20%. The predicted moduli are higher than experimental values for polymethacrylimide foams (Table 11.1). The face fractions are calculated from measured edge diameters, face thickness and cell diameters, and the measured Young’s polymer modulus was 6.9 GPa. Thus FEA of both irregular and regular foams, with unwrinkled faces, overestimates the foam Young’s modulus. Roberts and Garboczi (2001) created a Voronoi model for closedcell foams, in which the edges appear to have circular cross sections. However, the lowest relative density was 0.104, for which the fraction φ of polymer in the faces was 0.85. For this R, the predicted Young’s modulus was 10% higher than that given by equation (11.27).


Table 11.1

275

Shulmeister et al. (1998) predictions for Rohacell polymethacrylimide foams Foam density (kg m3) 38 116

Face fraction, φ

Measured foam modulus (MPa)

Predicted foam modulus (MPa)

0.59 0.49

20.1 83

43.3 100

0%

20%

40%

60%

Figure 11.17 CT reconstructions of bead foam structure after successive impacts to compressive strains of 0%, 20%, 40%, and 60% in the vertical direction (Viot and Bernard, 2005 © Kluwer).

11.8

Bead foams 11.8.1 Deformation mechanisms Viot and Bernard (2005) described the repeated impact of an expanded PP (EPP) of density 80 kg m3, in a rig that used a metal end stop to limit the compressive strain. Subsequent X-ray CT resolved the bead boundaries, but not all the cell walls, in a 2 mm cubic region inside the unloaded foam. Figure 11.17 shows that after impact to 20% strain the bead boundaries are somewhat distorted; after further impacts to 40% and 60% strain, some boundaries have become buckled. More material

276 Polymer Foams Handbook moved into the constant scanned volume as the impact strain increased. No attempt was made to estimate the volume of the bead boundary channels, and the amount of recovery was not stated.

11.8.2 Models for compression In bead foams, the air channels at the bead boundaries are usually a few per cent of the total volume. Some speciality products, such as Brock foam, deliberately increase the channel volume to about 20%, by bonding together PP beads with a PU binder. The web site for this material claims that it has significantly better shock absorbing properties than conventional foams. Consequently, a dynamic FEA model was made of foam beads, arranged in a BCC array (Mills and Gilchrist, 2007). In the simulation of [001] direction compression (Fig. 11.18), there is shear on the inter-bead junction that is oblique to the stress axis. The cusp of the air channel acts like a crack in a solid material in causing a stress concentration. The air channel volume slowly decreases as the foam strain increases (Fig. 11.18c). The predicted stress–strain curves, for bead foams with varying volume % of air channel, are compared with that for uniform PP foam of density 60 kg m3. The presence of the inter-bead channels significantly reduces the Young’s modulus, then reduces the stress over the rest of the stress–strain curve. Therefore, EPP foams with 20% air channel volume have a lower energy absorption capacity than the bead material, but there is no radical change in the material response. The mechanical response of most EPS, with 5% air channels, is almost the same as that of closed-cell PS foam without a bead structure.

11.9

Discussion For many commercially important closed-cell foams (LDPE, EPS) the air phase provides a major part of the compressive resistance. Analysis shows that in impact loading, heat transfer keeps the air effectively isothermal, while there is insufficient time for air loss by diffusion. Only under long-term creep loading is air loss a significant mechanism. The low Young’s moduli of PE foams, compared with predictions from a Kelvin foam model with flat faces, are likely to be due to face bowing; a result of contraction after production. The prediction from a Kelvin foam model, with the majority of the polymer in pre-buckled faces, is a lower bound on experimental Young’s modulus data. This model provides initial predictions of high strain compression response, in terms of the polymer modulus and tensile yield stress. It emphasises the high proportion of polymer in faces acting as membranes, and the effects of molecular and crystalline orientation on the face tensile yielding response. It is likely that FEA will eventually provide better models


0.5

0 0.4

3

277

0.1 0.3 0.2

2 1

(a)

(b) 1

0.25

Solid

0.2 WK 0.007 WK 0.085 OLS 0.21

0.6

0.15

0.4

0.1 OLS 0.21

0.2

0

Air channel volume fraction


0.8

0.05

0

0.1

0.2

(c)


0.6

0.7

0 0.8

Figure 11.18 Model of bead foam compression: (a) undeformed with loading plates; (b) at 40% compressive strain with stress contours (MPa); and (c) predicted EPP stress–strain graphs for different volume fractions of air channel, and air channel volume vs. strain (WK, wet Kelvin; OLS, overlapping sphere model) (Mills and Gilchrist, 2007).

of the interaction between cell gas compression and polymer deformation in these materials. There has been very little modelling of EPS or EPP foam responses, in spite of their commercial importance.

References Almanza O., Masso-Moreu Y. et al. (2004) Thermal expansion coefficient of polyethylene foams – theory and experiments, J. Polym. Sci. B Phys. 45, 3741–3749.

278 Polymer Foams Handbook Bureau M.N. & Gendron R. (2003) Mechanical–morphology relationships of PS foams, J. Cell. Plast. 39, 353–367. Bureau M.N., Champagne M.F. & Gendron R. (2005) Impact– compression–morphology relationships in polyolefin foams, J. Cell. Plast. 41, 73–85. Burgess G.J. (1988) Some thermodynamic observations on the mechanical properties of cushions, J. Cell. Plast. 24, 56–69. Chen C.P., Anderson W.B. & Lakes R.S. (1994) Relating the properties of foam to the properties of the solid from which it is made, Cell. Polym. 13, 16–32. Clutton E.Q. & Rice G.N. (1991) Structure property relationships in thermoplastic foams, Cellular Polymers Conference, RAPRA Technology, Shawbury, Shropshire, pp. 99–107. Gibson L.J. & Ashby M.F. (1988) Cellular Solids, Pergamon, Oxford. Kraynik A.M., Neilsen M.K. et al. (1999) Foam micromechanics, Foams and Emulsions, Eds. Sadoc J. & Rivier N., Kluwer, Dordrecht, pp. 259–286. Li Q.M., Mines R.A.W. & Birch R.S. (2000) The crush behaviour of Rohacell-51WF structural foam, Int. J. Solid. Struct. 37, 6321–6341. Loveridge P.L. & Mills N.J. (1991) The mechanism of recovery of impacted high-density polyethylene foam, Cell. Polym. 10, 393–405. Masso-Moreu Y. & Mills N.J. (2004) Rapid hydrostatic compression of low density polymeric foams, Polym. Test. 23, 313–322. McKown S. (2005) The progressive collapse of novel aluminium foam structures, Ph.D. thesis, University of Liverpool, Liverpool. Mills N.J. (1994) Impact properties, in Low Density Cellular Plastics, Eds. Hilyard N.C. & Cunningham A., Chapman and Hall, London. Mills N.J. (1996) Accident investigation of motorcycle helmets, Impact 5, 46–51. Mills N.J. (2005) Plastics: Microstructure and Engineering Applications, 3rd edn., Butterworth Heinemann, London. Mills N.J. & Gilchrist A. (1997a) Creep and recovery of polyolefin foams-deformation mechanisms, J. Cell. Plast. 33, 264–292. Mills N.J. & Gilchrist A. (1997b) The effects of heat transfer and Poisson’s ratio on the compressive response of closed-cell polymer foams, Cell. Polym. 16, 87–119. Mills N.J. & Gilchrist A. (2007) Properties of bonded polypropylene bead foams, J. Mater. Sci. (to appear). Mills N.J. & Zhu H.X. (1999) The high strain compression of closedcell polymer foams, J. Mech. Phys. Solid. 47, 669–695. Ostrogorsky A.G., Glicksman L.R. & Reitz D.W. (1986) Aging of polyurethane foams, Int. J. Heat Mass Trans. 29, 1169–1196. Renz R. & Ehrenstein G.W. (1982) Calculation of deformation of cellular plastics by the finite element method, Cell. Polym. 1, 5–13. Roberts A.R. & Garboczi E.J. (2001) Elastic moduli of model random 3-D closed-cell cellular solids, Acta Mater. 49, 189–197.


279

Rusch K.C. (1970) Load-compression behaviour of brittle foams, J. Appl. Polym. Sci. 14, 1263–1276. Santoza S. & Wierzibicki T. (1998) On the modelling of crush behaviour of a closed cell aluminium foam, J. Mech. Phys. Solid. 46, 645–669. Shulmeister V. (1998) Modelling of the mechanical properties of low-density foams, Ph.D. thesis, Technical University of Delft, Shaker Publishing, Maastricht. available at /repository.tudelft.nl Simone A.E. & Gibson L.J. (1998) The effects of cell face curvature and corrugations on stiffness and strength of metal foams, Acta Mater. 46, 3929–3935. Skochdopole R.E. & Rubens L.C. (1965) Physical property modification of low-density polyethylene foams, J. Cell. Plast. 1, 91–96. Throne J.L. & Progelhof T.C. (1984) Closed-cell foam behaviour under dynamic loading, Part I. Stress strain behaviour of low density foams, J. Cell. Plast. 21, 437. Viot P. & Bernard D. (2005) Impact test deformations of polypropylene foam samples followed by microtomography, J. Mater. Sci. 41, 1277–1279.

Chapter 12

Product packaging case study

Chapter contents 12.1 Introduction 12.2 Simple drops with the box parallel to the floor 12.3 Design of EPS mouldings 12.4 Other factors in packaging design 12.5 In-package tests of shock absorption Summary References

282 283 293 298 301 305 305


12.1

Introduction There is a wide range of foam packaging types for electronics or consumer goods, some of which are shown at www.sealedaircorp.com. They include: (a) Bead-foam mouldings with complex shapes (Fig. 12.8b). The usual polymers are expanded polystyrene (EPS), polypropylene (EPP), and polyethylene (PE) copolymers (Arcel). (b) Sections (hot wire cut from sheets of extruded PE foam) which may be assembled with adhesives into more complex shapes (Fig. 12.17). (c) Loose-fill foam shapes (peanuts, warped discs, and dumb-bells) made from EPS or biodegradable foamed starch (Fig. 12.1a). (d) Foam-in-place: the cardboard box and a simple former for the product shape are covered in a PE film, then polyurethane (PU) foam is injected into the gap. (e) Blocks of soft open-cell foam (Fig. 12.1b), cut while deformed by an array of finger-like projections, which can be on a pair of rotating rolls. When the foam is released, the cut surface forms an ‘egg-box’ pattern of hollows and protrusions. The flat side is usually bonded to cardboard. This provides extremely soft cushioning for delicate electronic circuits, etc. (f) Layers of close-packed cells in low-density polyethylene (LDPE) films. Bubblewrap sheet is made by welding two LDPE films together after bubbles of typically 10 mm diameter and 4 mm depth have been thermoformed. (g) Macro-bubbles, typically 150 mm long and 75 mm wide, made from lay-flat LDPE tubular film. The tube is inflated and then heat welded at intervals with serrated cuts in the weld regions, so that the bubbles can be separated. They are used either singly or connected end to end. This chapter concentrates on types (a) and (b) as the main types of heavy duty packaging. Takara and Cunningham (1998) reviewed the

40 mm (a)

Figure 12.1

30 mm (b)

(a) Loose-fill foam shapes and (b) convoluted flexible PU foam.

Chapter 12 Product packaging case study

283

mechanical properties of loose-fill foams but provided no impact data; their mean density is lower than type (a) with consequent lower energy absorption per unit volume. Closed-cell foams, by enclosing air, use the air as part of the cushioning. Consequently their compressive Young’s modulus should exceed 100 kPa (it may be slightly less for plasticised polyvinylchloride (PVC) foam, where Poisson’s ratio is high). Open-cell foams, that provide the same modulus, are usually denser. Types (f) and (g) are not strictly foams, but their geometry is similar. With large cells and few barrier films to hinder air diffusion, they have poorer creep resistance than foams and so cannot support heavy products in storage. They have limited form stability and so are used to fill the space between a product and a cardboard box. There is little information available about the impact performance of macrobubbles when tested in-box. Their response should be similar to that of soft, closed-cell foam, with an initial loading stage where touching bubbles deform in shape. Cardboard (corrugated fibreboard) is a type of two-dimensional foam, with cushioning properties similar to that of honeycombs, when the compressive stress is perpendicular to the axes of the cells. The thickness is usually between 3 and 8 mm, with densities the order of 100 kg m⫺3. Maltenfort (1998) discusses the design of cardboard boxes, while information is available at www.boxboard.com. Much early packaging design comes from US military sources (Mustin, 1968) and is related to airdrops of supplies. The shocks, to which packaged goods are exposed in the post, were measured with instrumented parcels (Society of Environmental Engineers, 1971). This chapter will consider in turn, simple rectangular blocks, truncated pyramidal shapes, and complex shapes of foam packaging.

12.2

Simple drops with the box parallel to the floor 12.2.1 Drop heights and fragility factors Calculation of product acceleration in a drop is the starting point for protective package design. During storage in a warehouse (Fig. 12.2a), the foam under the product is subjected to a static stress σs, related to the product mass m, the acceleration of gravity g, and the total foam cross-sectional area A by σs ⫽

mg A

(12.1)

The cardboard is assumed to take the majority of the loads from other boxes stacked above the one in question. If in subsequent handling the

284 Polymer Foams Handbook Cardboard Foam Product centre of mass

a Product

mg Total foam area A Rigid floor (a)

Rigid floor (b)

Figure 12.2 (a) Static loading of foam under a product in storage and (b) in a parallel-surface drop, the peak product acceleration a must not cause damage.

Table 12.1

Typical drop heights Type of handling

Weight (kg)

Drop height (m)

1 man throwing 1 man carrying 2 men carrying Hand trolley Medium handling equipment Fork lift truck

0–9 10–22 23–110 111–225 225–450 ⬎450

1.05 0.90 0.75 0.60 0.45 0.30

package is dropped, there is a risk of product damage. The parallelsurface impact configuration of Figure 12.2b is a worse-case scenario; when the package falls on a corner (Section 12.5.2), the peak acceleration of the contents is smaller. The analysis of parallel-surface impact is simple if the foam exists as discrete blocks below the product, since its geometry is the same as that of a uniaxial compression test. If the foam mouldings have complex shapes, methods of analysis are considered in Section 12.5. The drop height of the packaging is estimated from its size and the method of handling adopted. Table 12.1 contains data from Brown (1959) converted into metric units. The fragility factor of the product is a conservative estimate of the threshold linear acceleration level, measured in g, that causes mechanical damage. One g is the standard acceleration of gravity at the earth’s surface, 9.81 m s⫺2. Table 12.2 gives fragility factors for various types of electronic equipment. With the miniaturisation of equipment such as CD drives, the values have risen. The fragility factor concept assumes that product failure relates to peak acceleration, regardless of the acceleration vs. time shape. Some parts are damaged when they deform too much, and the deformation is a function of the shape of the acceleration–time history. In the US, shock tables are commonly used for evaluating packaging. This relatively expensive equipment, which may use

285


Table 12.2

Fragility factors Type of equipment

g

Gyroscopic instruments Computer monitors, printers Audio and television equipment, hard disk drives Household appliances and furniture Radiators, sewing machines, machine tools

Table 12.3

15–25 40–60 60–80 80–100 100–120

Cushion curve test methods Impact conditions

BS 4443

ASTM D1596-97

Sample dimensions (mm) 150 ⫻ 150 ⫻ 50 thick Conditioning Drop heights (m) Accuracy of peak, g (%) Number of impacts Time between (s) impacts Number of static stresses

At least 101 ⫻ 101 ⫻ 25 thick 23°C, 50% RH for 16 h Equilibrium: 23°C, 50% RH 0.25 and 0.75 Not specified 5 5 3 5 60 ⫾ 15 ⬎60 5

Not specified

pneumatic decelerators so that the package experiences an acceleration– time pulse with a constant central region, is based on military research (Newton, 1968). Such equipment has been criticised (Daum and Tustin, 2006) because the pulse shapes differ from real impacts; their introduction was a result of the deficiencies of shock instrumentation in the 1940s.

12.2.2 Cushion curve tests The website www.dow.com/perffoam/tech/design/index.htm introduces the cushion curve design method. Cushion curves relate the peak product acceleration to the foam thickness, the drop height, and the static stress σS. In a cushion curve test, the peak acceleration G, measured in g, of a vertically falling mass m is recorded after it falls a distance h onto a block of foam of thickness t. The test geometry is the same as that for uniaxial compression impact tests, shown in Figure 5.11. Table 12.3 compares details of BS 4443 Section 3, Method 9 Determination of dynamic cushioning performance with the equivalent American Society for Testing and Materials standard. In BS 4443, G is measured for three consecutive impacts on the same specimen. The ASTM measurement is either the average of G for the 2nd to 5th impacts on the same specimen, or G for a specific

286 Polymer Foams Handbook test of the series. Sequential impacts reduce the performance of most foams, so the test chosen should relate to the usage. In the BS tests, the static stress is varied by changing the falling mass. The range of static stresses is chosen by trial and error, so that the central value causes the minimum value of G, with the other stresses producing 10% and 20% increases in the peak acceleration. Ten samples are required to produce the cushion curve data. Results, for thickness other than 50 mm, can be obtained by testing or by computation. Each cushion curve is fitted through (G, σs) data for a particular foam thickness t, so a typical diagram shows a family of curves for a particular drop height h. For particular foams there will be a set of cushion curves for different drop heights.

12.2.3 Cushioning provided by the cardboard box Cardboard, made of high tensile strength wood fibres, has a much higher tensile strength than foam; so the box protects the foam from tensile loads. It also spreads localised loads in a similar manner to helmet shells. Sek and Kirkpatrick (1999) carried out compressive impact tests on 50 mm high stacks of cardboard layers. The value of the nearconstant collapse stress, in the impact stress–strain curve, depends on corrugated flute type (A to E) between the two flat skins and the humidity content of the cardboard. Values in the range 100–400 kPa are typical. Figure 12.3 shows some data and computed cushion curves for

Peak acceleration (g)

300

200

0.75 100

0 0

5

10

15

0.65

20

Static stress (kPa)

Figure 12.3

Predicted cushion curves for corrugated card flute B, thickness 50 mm, for drop heights (m) noted, compared with experimental data (individual points) Sek and Kirkpatrick (1997) © Wiley.


287

flute type B, with average length 6.5 mm and height 2.4 mm. Section 12.2.6 discusses the curve shape further. Naganathan and Kirkpatrick (1999) showed that cell air pressure can contribute to the impact response of cardboard. There is a similar air flow contribution to the impact response of some open-cell foams (Chapter 8). The density of cardboard is typically 5 times higher than the 20 kg m⫺3 of closed-cell foams which have a comparable compressive strength. Consequently it provides an additional thickness of crushable packaging.

12.2.4 Design using cushion curves The cushion curves, for the relevant foam and drop height (Fig. 12.4), are used as follows: • A horizontal line is drawn across at the level of the product fragility (e.g. 40 g). • The thinnest foam, for which the cushion curve passes below the line, is chosen (here, 75 mm of LD24 foam). • To minimise foam usage (cost), the design point on the curve is at 1.1 times the minimum g level in the direction of increasing static stress (here, 38 g at a static stress of 9 kPa). • The foam cross-section AF is calculated as the product mass divided by the static stress (for a 30 kg product, the area is 33 cm2). The mass of the foam, much less than that of the product, is ignored. Hence the impact kinetic energy is the same as for the product falling onto a foam-covered rigid surface. The cardboard box, needed 120 25 Peak acceleration (g)

100 80

50

60 75 40 100 20 0 0

Figure 12.4

5

10 15 Static stress (kPa)

20

25

Cushion curves for LDPE 24 foam, 1st drop from 0.75 m for foam thickness (mm) marked (redrawn from Zotefoams brochure).

288 Polymer Foams Handbook to enclose the product plus the foam may be custom made, or may be the smallest standard size available. A fraction φ of the lower area of the box needs to be covered with foam, where φ⫽

AF ABOX

(12.2)

There will be one design solution (AF, t) for each density of foam. The cheapest of these may be chosen or that which minimises the box size. Manufacturer’s cushion curve design programs, such as Ethacalc from Dow, assist in the calculations.

12.2.5 Calculating cushion curves from stress–strain responses Although the shape of the acceleration vs. time trace is recorded in cushion curve tests, no use is made of this information. However, analysis of one such trace can replace a large number of tests in determining cushion curves. Analysis can also reveal the maximum extent of crushing and the proportions of energy absorbed by the foam and returned to the rebounding mass. Grunebaum and Miltz (1983) tried to predict cushion curves from low-strain-rate compression curves for PU and urea–formaldehyde foams. However they did not consider the rate–dependence of foam properties. The experimental peak acceleration G, for a drop height of 0.6 m, was 30–50% higher than that calculated from the static stress–strain curve. Loveridge and Mills (1993) showed how to calculate cushion curves from a single impact test if the impact stress–strain behaviour of the foam follows a mastercurve. This occurs if the reduction in strain rate, when the strain approaches its maximum value, has a negligible effect on the stress. Mastercurves exist for all but the lowest-density polystyrene (PS) foams (Fig. 12.5), but not for LDPE foams, which exhibit marked viscoelastic responses. For a given drop height h, the kinetic energy of the product of mass m (Figure 12.2), at the moment of impact, is mgh. When the product has momentarily stopped, its kinetic energy is zero; so energy mgh has been input into foam blocks of initial volume At. Therefore the energy density U (J m⫺3) input into the foam is σm

U (σm ) ≡

∫

0

σ dε ⫽

mgh At

(12.3)

The integral U(σm) represents the area under the engineering stress–strain mastercurve up to the maximum stress σm. Combining


289

Stress (MPa)

3

2

1

0 0

Figure 12.5

0.2

0.4 0.6 Compressive strain

0.8

1

Compressive stress–strain mastercurve, for PS foam of density 56 kg m⫺3, is the envelope of the loading part of curves for impacts from a range of heights (Mills, 1994).

equations (12.1) and (12.3), the static stress is found to be a function of U(σm) σs ⫽

t U(σm ) h

(12.4)

The maximum product acceleration G in the laboratory reference frame occurs when the foam compressive stress has its maximum value σm. When the product is stored in its box, there is a static stress σs on the foam and the product acceleration is g in a freely falling reference frame. The ratio of the accelerations gives σ hσm G ⫽ m ⫽ g σS t U (σm )

(12.5)

The parameters σs and G of the cushion curve can be calculated from the energy density function U(σm), using equations (12.4) and (12.5). These equations only contain the ratio t/h of the foam dimensions, so the resulting cushion curves can be labelled with the reduced foam thickness ⫽ t/h. Figure 12.6 shows a set of predicted cushion curves for LDPE foam of density 70 kg m⫺3. The predictions are limited by the maximum stress in the impact stress–strain curve. Thus the curve for the 25 mm thickness foam ends at a static stress of 20 kPa because the impact stress–strain curve ended at a stress of 2.3 MPa. The correspondence between the predicted and the measured cushion curves for a 1st impact was found to be excellent.


Prediction 25

Maximum acceleration (g)

100

Experiment

80

60

40

50

20 100 0

Figure 12.6

0

4

8

12 16 Static stress (kPa)

20

24

Cushion curves for 1st 0.6 m drop on LDPE foam (thickness in mm) of density 70 kg m⫺3 : predictions (Mills, 1994) compared with Zotefoams data (solid curves).

Burgess (1990) confirmed this approach. He tabulated peak accelerations G and static stresses σs for impacts on LDPE copolymer foam of density 32 kg m⫺3. He showed that the data for σs h/t, which is the energy density input U from equation (12.4), and (G ⫹ 1)σs, which is σm from equation (12.5), can be interpolated graphically to estimate the maximum acceleration for any other static stress. He probably used G ⫹ 1 because the accelerometer output was zeroed before the drop commences, whereas Loveridge and Mills (1993) zeroed the accelerometer output while the mass was freely falling. Burgess (1994) also proposed that the cushion curves could be predicted from a single impact stress–strain curve, provided the strainrate dependence of the peak stress was relatively low. Inspection of Figure 12.7 for high-density polyethylene (HDPE) foam shows that the maximum stress in a particular impact falls about 10% below the curve for a very high-energy impact. Therefore, if this high-energy impact curve is as the mastercurve, the predicted G values will be about 10% too high. This is a conservative design method because the foam area will be 10% larger than strictly necessary. The design procedure can be carried on a computer (Zotefoams, 1991). The above analysis does not consider the unloading part of the impact stress–strain curve. The product rebounds because an energy density Uout is returned by the unloading foam.

291


Stress (MPa)

2

1

0

Figure 12.7

0

0.1

0.2


0.7

0.8

Compressive data for HDPE foam of density 115 kg m⫺3 impacted from a range of heights (Mills, 1994).

12.2.6 Cushion curves for specific stress–strain responses Attempts have been made to rationalise cushion curves by using a specific form for the compressive stress–strain curves for a class of foams. To illustrate this, an idealised, three-stage stress–strain curve is considered appropriate for cardboard; the foam is rigid (infinite Young’s modulus) until it yields at a constant compressive stress σ0 and until the strain is 1 ⫺ R; then the stress shoots up vertically as the foam bottoms out. When it is substituted in equations (12.3)–(12.5), the resulting cushion curve has a hyperbolic shape

G⫽

σ0 tσ if σs ⱕ 0 (1 ⫺ R) σs h

(12.6)

followed by a vertical increase when the foam bottoms out. The minimum in G occurs just before the foam bottoms out, so the foam area is chosen to give this value of static stress. Figure 12.3 shows just such cushion curves. Ruiz-Herrero et al. (2005) conflated Burgess’s (1994) analysis, in which a cushion curve is calculated from a single impact stress–strain curve, with his 1988 prediction of stress–strain curves, ignoring the polymer contribution and assuming either an adiabatic or isothermal air response (discussed further in Section 11.4.3). They made allowances for the finite relative density of the foam and the polymer contribution to the compressive yield stress. Thus, they started with a number of models for closed-cell foam compressive response. While Burgess (1988) predicted that the peak g at the minimum of a cushion curve was about 15% lower for his isothermal model than for his adiabatic model,

292 Polymer Foams Handbook Ruiz-Herraro et al. (2005) predict very similar cushion curves for the two cases. Their 2006 experimental cushion curves do not exactly match any of the models. Section 7.4.2 analysed the sound wave effects at the start of an impact compressive stress–strain response of a foam. There will also be strain rate effects, discussed in Section 19.5, close to the maximum strain of the compressive response. While it has been shown that the main features of a cushion curve derive from a single stress–strain curve, these two further effects will cause some deviations from the predicted cushion curves. Since cushion design is not a high precision exercise, given the uncertainties of real drop parameters, these deviations are of little importance.

12.2.7 Energy absorption diagrams and foam selection Maiti et al. (1984) integrated experimental compressive stress–strain data, as in equation (12.3), to calculate the input energy density U as a function of the maximum stress σm. They presented these data as an energy absorption diagram, in which U was plotted as a function of σm on log–log axes. Both parameters were normalised by dividing by the polymer Young’s modulus EP. However, this diagram does not have the direct design usefulness of cushion curves, as it neither incorporates the static stress nor indicates what parameters minimise the product peak acceleration. Normalisation of foam properties is not helpful because EP is not a separable constant in the response. Over the years, the LDPE used for foams has changed from ICI long-chain-branched polymer, to linear low-density PE, to metallocene PE. These differ in melt rheology, the degree of crystalline orientation in the cell walls, and crystallinity. The first factor can affect the foam geometry, while the last two affect the cell face Young’s modulus. As explained in Chapter 11, the compressive properties of PE foams depend on the cell face Young’s modulus, the tensile yield stress (which is only roughly proportional to the modulus), and the amount of cell face wrinkling after processing. Hence normalization by one of these factors is misleading. Nor is it helpful to link particular polymers to particular categories of foam response, such as elastomeric, plastic, and brittle. Some lowcrystallinity, low-density PE copolymer foams are almost elastomeric, but many others show plastic responses. Material selection is strongly influenced by factors other than the foam impact properties, which are a given starting point. For example, package designers tend to use EPS or LDPE for cost and processing reasons, while shoe sole designers use ethylene vinyl acetate (EVA) foams for processing and durability.


12.3

293

Design of EPS mouldings 12.3.1 Moulding geometry EPS packaging mouldings locate the product in the box and must stay in place during impact. The corners of the product are usually surrounded with ribs, while the foam is often cut away in the centre of the product faces. At the stage of mould opening, the moulding size has shrunk, causing it to grip the male side of the mould. Consequently the width of ribs (Fig. 12.8) is tapered to assist release from the mould. EPS moulding design is considered for two basic shapes, which are supported on a base in the moulding: (a) Tapered ribs: the length of the rib is many times its width, so the deformation is approximately plane strain in the plane perpendicular to the rib length. (b) Truncated pyramids, with a square base. A BASF (1992) brochure recommends that, when ribs are used, the cushion area is that at the mean rib height and the calculated cushion thickness is increased by 10%. The rib height should be 50–60% of the total foam thickness, the flank angle α should be about 15°, and the radii, where the rib meets the base, about 10 mm. Elsewhere BASF (1997) recommends that the EPS density should either be 18 kg m⫺3 for fragile or light goods or 24 kg m⫺3 for other goods. Use of these standardised densities simplifies the provision of design data.

12.3.2 Stupak and Donovan’s model for tapered ribs Stupak and Donovan (1992) assumed that there was a uniform y direction compressive stress in each layer of the tapered rib, the value of

Rib Area A d

H α Base

(a)

Figure 12.8

(b)

(a) Key dimensions of a tapered rib on a base and (b) a truncated pyramid section of a moulding.


Flat striker

Layer 5 Mirror symmetry plane

y Layer 3

Layer 1 Rigid flat base (a)

Figure 12.9

(b)

Principal stress directions in tapered ribs: (a) Stupak’s layer model and (b) FEA predictions in the upper part of a rib with taper angle 13° at a mean compressive strain of 20%. The line length is proportional to the stress (Masso-Moreu and Mills, 2002).

which decreases as the cross-sectional area increases (Fig. 12.9a). However, at the sides of the rib, the principal stresses must be parallel to the free surfaces; so the stress field must be more complex (Fig. 12.9b). In their series model, which was used by Masso-Moreu and Mills (2002), the same total force F acts on each layer; so the vertical stress on layer i was F/Ai. The uniaxial compression stress–strain response of the foam was described by equation (11.4) (σ0 ⫽ 352 kPa and p0 ⫽ 135 kPa for EPP of density 43 kg m⫺3). When inverted, this gives the vertical compressive strain εi in layer i as εi ⫽

σ ⫺ σ0 (1 ⫺ R) σi ⫺ σ0 ⫺ p0

(12.7)

The total rib deflection x is the sum of the layer deflections. The predicted responses (Fig. 12.10), for four taper angles, were good approximations to the loading part of the experimental data. The model ignores changes in the rib geometry as it is compressed. High-speed photos of the impact of a tapered rib of extruded polystyrene (XPS) foam (Gilchrist and Mills, 2001) showed that the upper part of the foam side surface moved into contact with the steel anvil. This was confirmed in more detailed photographs during slow compression tests (Masso-Moreu and Mills, 2003). Grids, with a 5 mm spacing, were marked on the end surface of tapered ribs. Superposed images, of the initial and deformed grids (Fig. 12.11a), show that the deformation spreads from the top of the rib, with no significant


295

3.5 Predictions 3 11.3

Force (kN)

2.5

18.4

2

1.5

25

Experiment 29.5

1

0.5

0

0

10

20

30 40 Deflection (mm)

50

60

Figure 12.10 Experimental data for impact compression of a tapered rib of EPP43 compared with the Stupak model predictions for the labelled taper angles (Masso-Moreu and Mills, 2002).

(a)

(b)

Figure 12.11 Deformed compared with undeformed grid:! (a) tapered rib with taper angle 29.5° at 53% mean strain and (b) truncated pyramid with taper angle 22° at 55% mean strain (Masso-Moreu and Mills, 2003).

deformation in the lower portion until the average strain is high. The upper, initially horizontal, grid lines have become circular arcs, ending at the upper anvil/foam interface. Lower, initially horizontal, grid lines have more complex shapes. Changes, in the angles between grid lines,


0.25 0.3

0.4

0.45

0.3

0.3

0.4 0.35 0.25 Mirror plane

(a)

Mirror symmetry

(b)

Figure 12.12 Truncated pyramid, of EPS of density 35 kg m⫺3 with taper angle 9.5°. Contours of von Mises equivalent stress in MPa at mean compressive strains of (a) 11% and (b) 39% (Mills, unpublished).

identify high shear strain regions near the upper outer corners of the foam. Grids on the truncated pyramid surface (Fig. 12.11b) show less deformation, indicating that the deformation is mainly internal.

12.3.3 FEA of truncated pyramids Masso-Moreu and Mills (2003) used a crushable foam material model (Section 6.3.2) in static finite element analysis (FEA), approximating truncated pyramids by axisymmetric truncated cones to achieve stable simulations. By 2006, it was possible to use dynamic FEA on truncated pyramids (author’s unpublished results). These were 75 mm high, had a base width of 75 mm, and top half-widths (in mm) (taper angles, °) of 10 (20.1), 15 (16.7), 20 (13.1), or 25 (9.5). Figure 12.12 shows the predicted von Mises stress contours; a high stress region runs obliquely from the upper corner of the pyramid in which plastic deformation is concentrated. At the mean strain of 39%, the contact area at the top of the pyramid has considerably increased. There is some similarity with the initiation and propagation of a neck in a PE tensile specimen. Figure 12.13 shows that the predicted force–deflection curves are close to the experimental impact data. However, there is more energy returned to the striker in the experiments than in the predictions. In the


297

2

Force (kN)

1.5

25

1

20 15 10

10

0.5 25

0

0

10

20

30

40

50

60

Deflection (mm)

Figure 12.13 Force–deflection curves for truncated pyramids of EPP of density 43 kg m⫺3, predicted by dynamic FEA (Mills, unpublished), for the labelled half-top widths (mm), compared with experimental graphs for 10 and 25 mm half-widths (dashed) for 1 m drops (Masso-Moreu and Mills, 2002).

former, expansion of compressed cell air returns energy, whereas in the latter the high Young’s modulus returns very little energy. The material model does not consider viscoelasticity. The experimental data for the 10 mm half-width pyramid just enter the ‘bottoming out’ region when the mean strain exceeds 80%. There are no signs of force oscillations due to the propagation of sound waves through the foam pyramids; this contrasts with the simulations of impacts on rectangular blocks of foam (Chapter 6).

12.3.4 Design rules for ribs Simple rules emerge from both the experiments and the FEA: 1. The initial collapse force is the product of the foam initial collapse stress and the pyramid top surface area. 2. The slope of the loading relationship is constant independent of the taper angle. An optimum shock absorber would provide a near-constant force; there is little to be gained by having a taper angle greater than the

298 Polymer Foams Handbook circa 10° value that allows easy mould release. Tapered ribs have a greater top surface area than truncated pyramids and, so should be more efficient. Given the requirement of minimising the size of the outer packaging box, it is not surprising that the taper angles on commercial foam blocks range from 8° to 15°, and that tapered ribs are more commonly used than truncated pyramids. FEA suggests that the BASF (1992) design hints, mentioned in Section 3.3.1, are valid. A comparison was made of the peak force vs. the input impact energy for both the uniaxial compression response of the XPS35 foam and the FEA model for a truncated cone with a 17.5° taper angle. The graphs were within 5% for deflections greater than 20 mm.

12.4

Other factors in packaging design Four further issues, mentioned in the Dow design guide, are explained below.

12.4.1 Foam creep during storage The creep stress on the foam, when the product is stored, is equal to the static stress. If this leads to a creep strain ⬎10% after 1000 h, the foam height will be less than 90% of the original, hence the performance in a drop will have deteriorated. Therefore the design static stresses should not cause excessive creep. Chapter 19 considers the creep of closed-cell foams.

12.4.2 Package vibration in transit The foam acts as an elastic spring with the spring constant k given by k⫽

EA t

(12.8)

where t is the thickness of the block. For a product of mass m, the resonance angular frequency ω of the package, when oscillating vertically, is given by ω⫽

k m

(12.9)

A low resonance frequency, relative to the forcing frequency, will reduce the transmission of vibration – see the discussion of car seat design in


Table 12.4

299

Frequencies experienced in transport Transport mode Aircraft floor Truck suspension Truck tyres Rail wagon suspension Rail wagon structure Ship deck

Typical frequencies, (Hz) 100–200 2–7 15–20 2–7 50–70 10

Chapter 9. However the resonant frequency should not be stimulated. Table 12.4 shows the frequencies typical of various types of transport. Heavy masses on soft packaging may resonate in some forms of transport. Edwards (1999) shows that vibration sensitivity is critical for products, such as hard disk drives, which act as a resonant system with a single degree of freedom. Electronic products can be mounted on a shaker table, subjected to sinusoidal oscillation, and tested for their acceleration sensitivity.

12.4.3 Temperature For military applications where low temperatures are experienced in high altitude flight and for high temperatures in tropical regions, foam performance over a wide temperature range is important. Even inside cars, the temperature range can be from ⫺40°C to nearly 100°C, depending on the country of use. The yield stress of PS foam reduces as it approaches the glass transition temperature, while the presence of residual blowing agent reduces the Tg of the polymer. Wyskida and McDaniel (1980) explored the change in cushion curves for EPS and PE foams as a function of temperature from ⫺54°C to 71°C. They describe mathematically the temperature dependence of the cushion curves, so that interpolation is possible between measurement temperatures. The yield stress of semi-crystalline PE decreases continuously in the temperature range between the Tg of ⫺120°C and the crystalline melting point of 110°C.

12.4.4 Multiple impacts Cushion curves can be generated for the nth member of a series of impacts, but this begs the question of whether a foam energy absorber is subjected in use to n identical impacts, with recovery times of 1 min. Although foam performance deteriorates after a single impact, only a small percentage of the foam in the packaging is affected.

300 Polymer Foams Handbook 89 30

82 91

Strain (%)

77 20

62 74

98 kg m−3

48 10

38

86 kg m−3

42 73 kg m−3 0 101

102

103

104

105

106

Time (s)

Figure 12.14 Strain recovery at 23°C, after impacts of HDPE foam of density shown, to peak strain % values indicated by the start of each curve (Mills and Hwang, 1989).

Impacts on the same site on a package are infrequent. However, in some applications repeated impacts occur on the same site, for example head protectors for boxers and knee-pads for skate boarders. Some foams (LDPE, HDPE, PP) recover viscoelastically after an impact; so their performance in a second impact depends on the length of recovery time, while others (PS) do not. Recovery of the original foam shape should not be confused with the regain of the original stress–strain curve. The latter requires the complete recovery of foam microstructural geometry; cell edge or face damage by yielding or fracture weakens the foam in a second impact. Mills and Hwang (1989) investigated the performance of HDPE foams after several impacts. Dimensional recovery, after a high-strain impact, took the order of a week (Fig. 12.14), but could be accelerated if the foam were heated to 50°C. Totten et al. (1990) carried out similar tests on a range of relatively soft LDPE and polypropylene (PP) packaging foams, with densities in the range 30–35 kg m⫺3. These were impacted up to 15 times with 1 min intervals between the impacts. After the first impact, the initial ‘yield’ point was lost due to permanent deformation in the polymer. Unless the foam faces fracture, the pneumatic contribution of these closed-cell foams stays constant. Figure 12.15 shows the cushion curves for the 1st, 5th, and 15th impacts of PP foam of density 30 kg m⫺3. The minimum of the 1st impact cushion curve is at a static stress of 8.3 kPa, but at a lower stress for multiple impacted foams. Mills (1994) investigated multiple impacts on EPS of density 56 kg m⫺3. There was no detectable recovery (measurements started 10 s after the impact) and a series of impact tests at 60 s intervals


301

300

Peak acceleration (g)

15

5

200

1 100

0

4

6

8 10 12 Static stress (kPa)

14

16

Figure 12.15 Cushion curves for the 1st, 5th, and 15th drops of a PP foam of density 30 kg m⫺3 from 0.9 m (redrawn from Totten et al., 1990 © Wiley).

shows a marked change in the stress–strain curve. The strains were calculated from the foam thickness prior to the first test. The second loading curve goes through the (peak stress, peak strain) co-ordinates of the first impact curve, and so on. After 1 h of recovery at 50°C the initial yield point has returned, but at a lower stress than in the first test, showing permanent impact damage. For this high yield-stress foam, the compressed air in the cells is ineffective in restoring the initial structure. For strains ⬎emax of the previous impact, the curve follows the master curve for high-energy impacts. This implies that the damage to PS foam is a unique function of the maximum strain to which the foam has been subjected. This is used in Chapter 16 for forensic inspection of EPS helmet liners.

12.5

In-package tests of shock absorption 12.5.1 Introduction Cushioning types (d) to (f), mentioned in the introduction, can only be tested with a package. During in-package tests, there may be extra energy absorption mechanisms: (a) crushing of the corrugated cardboard, (b) prevention of low-energy foam deformation modes, (c) pneumatic effects in the air spaces.

302 Polymer Foams Handbook However these contributions appear to be minor, as FEA models ignore them yet make good predictions of the package performance. Gorman (1997) discussed methods of performing in-package impacts. If the foam properties are affected by repeated impacts, the data used for in-package impact design must be for the appropriate number of impacts. If the foam moulding is sheared, it can crack, and this can reduce the performance in subsequent impact. Chapter 15 examines the cracking of EPS mouldings.

12.5.2 Corner and edge impacts Mustin (1968) analysed the effect of a wooden packing case falling on one corner or edge rather than flat on its base. The impact force F has a moment M about the centre of mass of the product. The rota.. tional acceleration θ x of the package is linked to its angular inertia Ix and to the moment (all quantities being about the x-axis) by Mx ⫽ I x θx

(12.10)

Mustin considered the equations for the motion of the product and container masses. However, the mass of cardboard boxes can usually be neglected compared with the product mass, so the simpler equation (12.10) could be used. The moment rotates the case until the lower surface is nearly parallel to the ground. During this stage, the energy absorbed by the foam is greater than the potential energy gained as the product centre of gravity falls. Consequently less kinetic energy remains for the eventual ‘flat’ impact. Dynamic FEA can predict the motions of a dropped box. A box, of geometry shown in Figure 12.1, was rotated by 10° about the x and y axes and then dropped onto the corner. The product mass was 4 kg, the impact velocity 7 ms⫺1, and the 50 mm cubes of EPS foam had density 35 kg m⫺3. The predicted variations of the vertical floor reaction forces with time, for the corner and flat drops, are compared in Figure 12.16a. The impulse for the corner drop is spread over a longer time, with a lower peak force, as predicted by Mustin. This also shows the need for in-box tests to be guided to obtain flat impact conditions. The box rotates after the initial impact until all four EPS blocks are almost equally compressed (Figs. 2.16b and c). FEA predicts vibrations in the ‘product’, which is a uniform block of material. When sound waves reflect from the corners of the product, there are locally high accelerations. Consequently the mean product acceleration does not necessarily describe the acceleration at particular locations. For real products it is possible to compute the loads on the critical components and their deformation (Low et al., 2001).


303

5 Flat drop

Impact force on box (kN)

4

Corner drop 10°, 10°

3

2

1

0

0

2

4

6

(a)

8 10 Time (ms)

12

0.1

14

16

0.1 0.2

0.2

0.3 0.3 3 2

(b)

1

0.4

Fri Jun

3 2

Fr

G

1

(c)

Figure 12.16 (a) Predicted variation in floor reaction force with time and box orientation after (b) 6 ms and (c) 12 ms. Contours of von Mises stress in the EPS foam and in the product (MPa) (Mills, unpublished).

12.5.3 FEA of in-box PE foam packaging Mills and Masso-Moreu (2004) considered the performance of some common foam block shapes in drop tests in a cardboard box. The ‘product’ was designed to be as rigid as possible, with resonant frequency ⬎2 kHz, to avoid its ‘ringing’ affecting the linear accelerometer signal. Steel blocks provided the main mass surrounded by a box constructed from aluminium honeycomb panels. The compressive stress–strain curve of Ethafoam E220 PE foam was fitted with the parameters R ⫽ 0.040, σ0 ⫽ 72 kPa, p0 ⫽ 75 kPa in equation (11.4). For end-cap foam designs, shown in Figure 12.17, the foam is mainly loaded in compression during the impact. The FEA gave good predictions of the force vs. package deflection. There were very small


Impact force on product (kN)

6

4 3 2 1 0

(a)

FEA 1st impact 3rd impact 5th impact

5

0

5

10

15

20

25

30

Package deformation (mm)

(b)

Figure 12.17 End-cap design (a) product and PE foam, with cardboard box removed and (b) predicted and measured impact force vs. foam deflection (Mills and MassoMoreu, 2004).

Mirror 4 plane

−50 0 0 50

50 2 3 1

100

150 4

50

Impact force on product (kN)

5

100

FEA 1st impact 3rd impact 5th impact

4

3

2

1

0 0

(a)

(b)

5

10

15

20

25

30

Package deformation (mm)

Figure 12.18 (a) Lower corner caps at nearly peak deflection and (b) FEA compared with measured impact responses (Mills and Masso-Moreu, 2004).

dynamic effects of the 5.6 kg product mass oscillating on the foam, so the assumptions of the cushion curve approach are validated. For the corner-cap design shown in Figure 12.18, the definition of static stress is unclear since the stress distribution in the foam under the product, due to static loading, is non-uniform. During a drop test there is a complex stress field in the foam with a stress concentration at the sides of the product corner. The FEA underestimated the experimental forces at a particular package deformation due to the material model for the LDPE foam being inadequate in tension. The model


305

assumes that the foam yields in tension at a low stress, whereas in reality the foam fails in tension at a stress that is 4 times the initial compressive yield stress.

Summary The cushion curve method, although widely used, is only an approximation, which works best if the foam is in the form of rectangular blocks in the package, and the drop is onto the package face. FEA is now capable of predicting the response of complex-shaped EPS or EPP mouldings used in packaging, and for corner or edge drops. Nevertheless, the FEA material models do not accurately simulate the tensile response of foams.

References BASF (1992) Brochure, Packaging with Styropor, HSR 8706, Ludwigshafen, Germany. BASF (1997) Brochure, Technical Information Styropor, Ludwigshafen, Germany. Brown K. (1959) Package Design Engineering, Wiley, New York. Burgess G.J. (1988) Some thermodynamic observations on the mechanical properties of cushions, J. Cell. Plast. 24, 56–69. Burgess G.J. (1990) Consolidation of cushion curves, Pack. Tech. Sci. 3, 189–194. Burgess G.J. (1994) Generation of cushion curves from one shock pulse, Pack. Tech. Sci. 7, 169–173. Daum M. & Tustin W. (2006) Drop tests vs. shock table transportation tests, on www. vibrationandshock.com Edwards (1999) FEA of the shock response and head slap behaviour of a hard disc drive. IEEE Trans. Magnet. 35, 863–869. Gilchrist A. & Mills N.J. (2001) Impact deformation of rigid polymeric foams: experiments and FEA modelling, Int. J. Impact Eng. 25, 767–786. Gorman S.P. (1997) In-package methods improve shock vibration testing, Pack. Tech. Eng. March and April issues. Grunebaum G. & Miltz J. (1983) Static versus dynamic evaluation of cushioning properties of plastics foams, J. Appl. Poly. Sci. 28, 135–143. Loveridge P. & Mills N.J. (1993) Prediction of packaging cushion curves and helmet liner responses, Cellular Polymers II Conference, RAPRA Technology, Paper 21. Low K.H., Yang A. et al. (2001) Initial study on the drop impact behaviour of mini Hi-Fi audio products, Adv. Eng. Software 32, 683–693.

306 Polymer Foams Handbook Maiti S.K., Gibson L.J. & Ashby M.J. (1984) Deformation and energy absorption diagrams for cellular solids, Acta. Metall. 32, 1963–1975. Maltenfort G.G. (1998) Corrugated Shipping Containers: An Engineering Approach. Jelmar Publication Co, Plainview, NJ. Masso-Moreu Y. & Mills N.J. (2002) Impact compression of polystyrene foam pyramids, WorldPak 2002, Proceedings of 13th APRI Conference on Packaging, Ed. Twede D, CRC Press, Boca Raton, FL, pp. 797–815. Masso-Moreu Y. & Mills N.J. (2003) Impact compression of polystyrene foam pyramids, Int. J. Impact Eng. 28, 653–676. Mills N.J. (1994) Impact response, in Low Density Cellular Plastics, Eds. Hilyard N.C. &. Cunningham A, Chapman & Hall, London, pp. 270–318. Mills N.J. & Hwang A.M. (1989) The multi-impact performance of high-density polyethylene foam, Cell. Polym. 8, 259–276. Mills N.J. & Masso-Moreu Y. (2005) Finite Element Analysis (FEA) applied to polyethylene foam cushions in package drop tests, Pack. Tech. Sci. 18, 29–38. Mustin G.S. (1968) Theory and Practise of Cushion Design, Shock and Vibration Information Center, US Department of Defense. Naganathan P., He J.M. & Kirkpatrick J. (1999) The effect of the compression of enclosed air on the cushioning properties of corrugated fibreboard, Pack. Tech. Sci. 12, 81–91. Newton R.E. (1968) Fragility Assessment: Theory and Test Procedure, US Navy Postgraduate School, Monterey. Ruiz-Herrero J.L., Rodriguez-Perez, M.A. & de Saja J.A. (2005/2006) Prediction of cushion curves for closed cell polyethylene-based foams, Part I. modelling, Cell Polym. 24, 329–346; Part II. experiments, ibid 25, 159–175. Sek M.A. & Kirkpatrick J. (1999) Prediction of the cushioning properties of corrugated fibreboard from static and quasi-dynamic compression data, Pack. Tech. Sci. 10, 87–94. Society of Environmental Engineers (1971) Shock Environment of Packages in Transit, SEE, London. Stupak P.R. & Donovan J.A. (1992) Geometry, load spreading and polymeric foam energy absorber design, SAE Trans. 101, Section 6, 263–269. Takara P.D. & Cunningham R.L. (1998) Properties of protective loose-fill foams, J. Appl. Polym. Sci. 67, 1157–1176. Totten T.L., Burgess G.J. & Singh S.P. (1990) The effects of multiple impacts on the cushioning properties of closed cell foams, Pack. Tech. Sci. 3, 117–122. Wyskida R.M. & McDaniel D.M. (1980) Modelling of Cushion Systems, Gordon & Breach, New York. Zotefoams (1991) Design Package, Zotefoams, Mitcham, Surrey.

Chapter 13

Running shoe case study

Chapter contents 13.1 Introduction 13.2 Foam selection and properties 13.3 Running biomechanics 13.4 Stress analysis of the foam in the shoe 13.5 Durability 13.6 Discussion References

308 309 313 317 321 326 326


13.1

Introduction 13.1.1 Information sources The major shoe manufacturers have web sites, which show their design innovations, for example www. Adidas (asicstiger/ Nikebiz/ reebok/ saucony).com. There is information on the history of trainers, and background about the materials, at www.sneakers.pair.com. This chapter considers the mechanics of foam cushioning, attenuating the forces of heel strike, in running shoes. Books by Frederick (1984), Nigg (1986), and Segesser and Pforringer (1989) consider the biomechanics of running shoes. The magazine Biomechanics, part of which is on the web at www.biomech.com, considers both orthotic and running shoe issues. International Society of Biomechanics, Technical Group on Footwear Biomechanics, organises biennial conferences on Footwear Biomechanics. Abstracts from the meetings are available at www.staffs.ac.uk/isb-fw. Orthotic foam inserts in shoes, used to correct a variety of walking and running problems, are considered by Hunter et al. (1995). An orthosis is an external device, which applies force to part of the body, to correct some problem.

13.1.2 Foam shoe components Shoe mass is important, since extra mass will add to the energy consumption of the athlete. Elite athletes use lightweight shoes (0.1 kg) for races, but heavier, more durable shoes for training. We will concentrate on trainers, which weigh about 0.3 kg. Figure 13.1 shows the component parts: (a) (b) (c) (d) (e)

Rubber outsole Foam midsole foam Foam innersole with cloth cover Heel counter Top cloth.

An old pair of trainers can be cut in half with a band-saw, to reveal the thickness of the various components. The midsole foam moulding is wedge shaped and runs the whole length of the shoe (Fig. 13.1), while traditional shoes would have a separate heel block.

13.1.3 Patented features in shoes Sales appeal depends on brand names, and on the external shape and colours. The external appearance and even tread pattern have distinctive designs, that emphasise real or imaginary internal components,

Chapter 13 Running shoe case study

309

PU foam in tongue and heel lining

Rubber outsole

Figure 13.1

EVA foam midsole

PU foam innersole

EVA foam inner sock

Cloth Foam insert Heel lining in heel counter

Section of a used running shoe showing the main components.

which provide cushioning or control torsion of the foot. The majority of the cushioning is due to the ethylene/vinyl acetate (EVA) midsole foam, some of which may be replaced by features such as: (a) Nike’s large bubbles of ‘air’ (Fig. 13.2) which apparently contain SF6 gas, as an insert in the main foam (see www.sneakers.pair.com/ airtech.htm and US patent 4219945). The polyurethane (PU) rubber capsule prevents the SF6 gas from diffusing out from the chamber at any significant rate, but air will diffuse in after the shoes are used, to restore the initial overall gas pressure. (b) Asics’s gel insert in the heel; its small size suggests that it plays a minor role in the shock absorption performance. (c) Puma’s thermoplastic honeycomb in the heel region. (d) Nike’s PU rubber springs in ‘Shox’ shoes.

13.2

Foam selection and properties 13.2.1 Material selection The majority of running shoes (trainers) have a midsole compression moulded from EVA copolymer foam. Polyvinyl acetate has the — O—CH3)]n . The vinyl acetate (VA) structure —[CH2—CH(—O—C — monomer units are randomly mixed with ethylene monomers in the copolymer sequence, reducing the crystallinity. The typical 18% VA content reduces the crystallinity of the EVA copolymer to about 20%.


(a) PU foam midsole

Gas chamber PU-shaped springs

Outsole

Foam

(b)

PU rubber hollow cylinder spring

Figure 13.2

Thermoplastic mouldings

Carbon filled rubber outsole

Longitudinal section through the heel region of shoes showing the materials used: (a) Nike Air and (b) Nike Shox.

This is far less than the 40% in low-density polyethylene, so the Young’s modulus of the polymer is very low, and the crystal melting temperature is about 70°C. Shoe manufacturers had steam-heated presses, for compression moulding rubber outsoles, which could be adapted to compression mould foamed EVA, which crosslinks at similar temperatures to synthetic rubbers. Reasons for the use of a density 200 ⫾ 100 kg m⫺3 emerge during the case study. The density can be increased to 300 kg m⫺3 in regions intended to give extra support. The white colour of basic material allows the incorporation of pigments, hence the multicolour mouldings of the exterior of trainers. The polymer itself is colourless, but the air bubbles strongly scatter light, providing the white appearance. The durability of the foam is


(a)

Figure 13.3

311

(b)

SEM of EVA foam from running shoes: (a) in centre of midsole and (b) near lower surface (on right).The sole thickness direction is horizontal and length direction is vertical (Verdejo and Mills, 2004a).

not as great as solid rubber components, but the sides of shoes suffer less wear than the soles. As shoe components must remain bonded together in use, it helps if the polymer has several mechanisms of adhesive action. Crosslinking reactions in the EVA form covalent bonds to the surfaces of other components. EVA copolymers are widely used as hot-melt adhesives because they contain polar groups and the melting point is low. The initially thermoplastic EVA can flow between the fibres of leather, before it crosslinks. Hence the melt viscosity of the polymer affects the bonding. As the molecular weight of the polymer increases, the melt viscosity increases rapidly, so melt flow into small gaps becomes more difficult. Adhesive bonding is cheaper than the alternative of stitching, as a method of securing shoe components.

13.2.2 Microstructure and processing of EVA foam In earlier shoe constructions, wedges were cut from EVA foam sheet and adhesively bonded to the outsole and shoe upper. EVA is often foamed in a two-stage process (Sims and Khunniteekool, 1996). A mixture of EVA polymer, blowing agent, and crosslinking system are compounded at 100°C, without foaming or crosslinking. When the compound is extruded as sheet, the chemical blowing agent decomposes generating a foam, which is cooled. In a second stage, sections are compression moulded at a higher temperature with the other shoe components and allowed to crosslink. The product is referred to by Nike and some other manufacturers as ‘Phylon’. The microstructure of running shoe EVA foam is shown in Figure 13.3. The closed-cell foam has the majority of the material in the cell

312 Polymer Foams Handbook faces, that is φ ⬎ 0.7 in the nomenclature of Chapter 11. The cell faces are relatively thick, compared with low-density polyolefin foams. There is little sign of connected vertexes forming stiff structures. Near the sole surface the cells are flattened.

13.2.3 Alternatives to EVA foam Promotional articles for novel polymer foams tend to give the impression that these materials are commercial important. Thus Pillow (1997) described the use of constrained geometry catalyst technology to make narrow molecular weight distribution ethylene–octene copolymers (87:13 molar ratio), which have 13% crystallinity and start melting at 42°C. They can be crosslinked using peroxides and foamed. Diegritz (1998) described the use of Dupont Dow Engage ethylene– octene copolymer foams as replacements for EVA shoe midsoles. The material can be injection moulded, which improves the economics of manufacture. It has better thermal stability than EVA, so the processing window is wider. After loading for 105 cycles, the residual deflection is 56%, less than the 70% for the EVA foam, showing that less creep has taken place. The foam is claimed to have a lower hysteresis than EVA foam, so it should not heat up as much. The main commercial alternative to EVA is PU foam. Indesteege et al. (1997) discuss the use of water-blown PU foam for midsoles (Table 13.1). The PU foam density is higher but the compression set (see Section 5.6.4) is much lower, presumably due to the lower permeability of the polymer to air. The extra shoe mass is a disadvantage. PU midsoles are more common in basketball shoes, where there are greater landing velocities. They also bond better to PU air bladders. Sawai et al. (2000) discuss the development of lower-density PU foams (circa 250 kg m⫺3) for use in sports shoes, and discuss how the split tear strength increases with foam density. The split tear strength measures the force required to continue a crack in a foam, after a split has been started, divided by the thickness of the foam.

Table 13.1

Comparison of PU and EVA foams for midsoles Test

EVA

PU

Density (kg m⫺3) Split tear strength (kN m⫺1) Compression set (60°C/6 h) Asker C surface hardness Pendulum rebound resiliency (%)

200 4.5 58 62 35

320 3.5 5 60 30

The Asker C hardness test uses a 5 mm diameter hemisphere pressed into the foam with a force of a few newtons.


313

Alley and Nichols (1999) differentiated between injection-moulded EVA (IP) of density 170–220 kg m⫺3 and compression-moulded EVA (Phylon) of density 200–250 kg m⫺3. There is little difference in terms of the properties in Table 13.1. However, the waste from compression moulding is ⬎50%, whereas it is ⬍10% for IP, which makes the average cost per Phylon midsole $1.80–2 compared with $1.2–1.4 for IP. For PU foam the cost is $1.55–1.65, since the tooling cost is less. Between 1996 and 1998 EVA gained at the expense of PU, with the 1998 data being 54% Phylon, 13% IP, and 33% PU.

13.2.4 Grades of foam Hardness measurements are used for quality control of foams to check the compressive modulus. In the Shore A hardness test, a steel pin of diameter about 1 mm is pressed into the material surface by a spring; the greater the elastic deflection of the material, the lower is the hardness. Different hardness measures only correlate with Young’s modulus for a particular material of a particular thickness. As there is a complex strain field under the indenter, and the foam has a nonlinear response in compression, the hardness is affected by other material properties than the Young’s modulus. Nike’s US patent 4535553 (1983) comments that EVA midsoles must have a Shore A hardness ⬎25, otherwise the foam will bottom out.

13.3

Running biomechanics 13.3.1 Midsole foam pressure distribution The main loads on the midsole are as follows: (a) Localised compressive loads from the heel, metatarsal heads, and hallux (big toe). The midsole cushions the impact forces in these regions. (b) Bending loads during part of the foot motion. Shorten (1993) mapped the pressure distributions in running shoe midsoles showing peak pressures in the order of 400 kPa in the heel region. Hennig and Milani (1995) tested 20 types of shoes, with athletes running at a marathon pace of 3.3 m s⫺1. For some shoes the pressure reached 1 MPa, but typically it was the order of 500 kPa. An animation of one test is viewable at www.uni-essen.de/⬃qpd800/ research.html. Figure 13.4a shows Verdejo and Mills (2002) measurements for a runner on a treadmill, wearing new shoes with EVA midsoles. The peak pressure, 280 kPa, is somewhat lower than in earlier research, but the treadmill surface is more compliant than a

314 Polymer Foams Handbook 10 min running

(a)

(b)

Figure 13.4

0

21

42

63

Calcaneus

83 104 125 146 167 188 203 229 250

Metatarsal

Hallux

(a) Map of peak pressure at each part of the shoe during running (Verdejo and Mills, 2002). (b) Side view of protuberances on the calcaneus, metatarsals, and hallux, which cause pressure peaks.

road surface. Tekscan (www.tekscan.com/medical) F-Scan technology was used to measure the pressure distribution.

13.3.2 Foot strike forces High-speed photography of runners shows that the average vertical velocity of the foot, when it strikes the ground is 0.5–0.7 m s⫺1. The force vs. time on the shoe, measured by a force plate, is shown in Figure 13.5a for a heel strike runner. The force, in the direction of motion, is much less. Swigart et al. (1993) split the vertical force component vs. time trace into two peaks: the initial impact peak and the subsequent active peak where leg muscles push the body from the ground. Bobbert et al. (1992) used motion markers on runner’s bodies, and modelled the running action as the motion of nine linked body segments. They showed that the initial impact peak is due to the support leg segments colliding with the floor; the lower leg does not rotate in


315

Heel tissue 3

F

Force (kN)

F2

Foam insole

2 F1

F Foam midsole

1

F Rubber outsole

0 (a)

Figure 13.5

50

150 100 Time (ms)

F

200 (b)

Ground

(a) Schematic of normal force vs. time for heel striker runners and (b) series loading of the heel and shoe components during heel strike.

this period, and the vertical knee and ankle accelerations are the same. The upper leg rotates, limiting the vertical acceleration of the hip. For the rest of the ground contact, between 40 and 100 ms, rotation of the lower leg limits the vertical forces on the foot. The upper and lower leg muscles do work on both landing and take-off, so the majority of the energy in the foot strike is consumed. The energy absorbed by the shoe foam and the heelpad is minor. Figure 13.5b shows that the force from the running surface passes in turn through the shoe components and the fat pad of the heel. Concrete or tarmac hardly deforms during running, while the shoe components and heelpad are deformable. This series arrangement means that the same force F is experienced by the heelpad, the shoe, and the ground, while the deflections are added. The shoe must be at least as compliant as the other two bodies to cushion the impact. For running on soft grass, there is no need for shoe cushioning, but on a road surface, the heel strike must be cushioned.

13.3.3 Foam flexure and heel stability The bending stiffness of the sole affects the runner. Figure 13.6a shows the flexure of a midsole, which tapers in thickness towards the front of the shoe. The midsole curvature depends on the local application of forces from the metatarsal heads. If there is a deliberate hinge in the outsole, the curvature may be localised at this point. However, the local compression of the foam by the metatarsals makes flexure easier, since the second moment of area of the cross-section is proportional to the thickness cubed. Consequently measurements of the

316 Polymer Foams Handbook (a)

(b) Force

Foam

Figure 13.6

Heel wear

(a) Midsole bending during push-off with tensile stresses near the lower surface and (b) uneven heel compression, with the heelpad indenting the foam, during heel strike.

bending stiffness of the sole without any compressive load are irrelevant to the shoe mechanics. The consequences of excessive sole bending stiffness could be strains in the tendons under the foot arch, which attach to the bone, causing planar fascitis. Energy loss mechanisms stabilise the ‘wobbling’ that could occur in shoe compression. During the compression of EVA foam there is energy loss, so only a fraction of the input energy is available to cause deformation elsewhere in the foam. However in a system without energy loss, instability is easier. Figure 13.6b shows the large mass of the runner, on the foam shoe sole. If the upper surface of the heel tilts, with associated pronation or supination of the talus/calcaneus joint, the runner’s potential energy will fall. The indentation of the foam in the heel area, during a heel strike, partly stabilises the situation. Excessively thick, soft shoe heels may allow excessive rotation of the ankle joint (pronation) and hence injury.

13.3.4 The effect of the shoe on running style The greater cushioning afforded by EVA foams allowed many distance runners to change their running style, from the forefoot striking the road first, to the heel striking first. Repeated heel strikes could damage joints if there were no cushioning in the shoes. Robbins and Waked (1997a) argued that deceptive advertising (that the shoes can cushion against impact) creates a false sense of security among the users of athletic shoes; if they modify their running style, they may suffer more injuries. In experiments, barefoot volunteers stepped down by 45 onto 25 mm thick layers of EVA foam, after being given positive, neutral, or warning messages about the foam properties. The peak force in the foot impact was affected by the message. In a related experiment (Robbins and Waked, 1997b), volunteers stepped onto EVA layers of different stiffness. The subjects tended to land hardest


317

on the softest foam, possibly because they aimed to compress it to make the landing more stable. Shorten (2000) argued that the runner’s body acclimatises to the mechanical loads during training. A sudden change to a lighter, less cushioned shoe for a race constitutes a training error, which increases the likelihood of injury.

13.4

Stress analysis of the foam in the shoe In this section, a typical foot strike is considered. Different levels of analysis are possible.

13.4.1 Uniaxial compressive response of EVA foam Figure 13.7 shows a compressive stress–strain curve, at impact strain rates, for EVA foam from a running shoe. The response changes quite rapidly with time from the initial state and appears to approach an equilibrium response. There is hysteresis on unloading. Misevich and Cavanagh (1984) modelled the foam response, assuming that the shoe prevents its lateral expansion, and that the compressive strain was uniform. The air in the EVA foam cells, which makes up 80% of the volume, was assumed to compress adiabatically. The cell air was taken to be an ideal gas, with the parameter γ ⫽ 1.4. They stated that heat flow, to the cell walls, contributed to the hysteresis energy loss. As no heat flow calculations were given, it is suspected that heat flow was not modelled. As the EVA has a low Young’s modulus, and the cell faces were thin, the polymer contribution to the stress was neglected. The impact of a striker on the foam was modelled at a series of small time increments. When the initial gas pressure p0 ⫽ 100 kPa, the predicted peak accelerations of the 8.5 kg striker were too high. Rather than considering a polymer contribution to the compressive stress, they repeated the simulation with an absolute gas pressure p0 ⫽ 169 kPa and obtained better prediction. To justify this high gas pressure in the undeformed foam they argued (unconvincing) that gas bubbles are released when the foam is cut under water. Equation (11.21) gives the cell gas temperature rise under adiabatic conditions. When the foam Poisson’s ratio and the polymer contributions are zero, the foam compressive stress becomes ⎛ 1 ⫺ R ⎞⎟γ ⎟ ⫺ pa σ ⫽ pa ⎜⎜ ⎜⎝ 1 ⫺ ε ⫺ R ⎟⎟⎠

(13.1)


0 min 30 min 60 min 90 min 120 min

450

Stress (kPa)

400 350 300 250 200 150 100 50 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Strain

Figure 13.7

EVA shoe foam impact stress–strain responses at various times during repeated impacts at 1.5 Hz (Verdejo and Mills, 2004b). 350 300 Stress (kPa)

250

Foam response

200 EVA structure

150 100 50

Isothermal air

0 −50

Figure 13.8

0

0.1

0.2


0.6

0.7

Impact compressive stress–strain curve for EVA foam of density 65 kg m⫺3, with the contributions from the isothermally compressed air and the EVA (adapted from Torden, 1993).

Torden (1993) measured the impact compression response of a foam described as EVA65; this was a crosslinked EVA foam, produced by the Zotefoams process, with density 65 kg m⫺3, so of lower density than shoe foams. He computed the isothermal response of the cell air. When this was subtracted from the foam response (Fig. 13.8), there appeared to be a constant polymer contribution of 140 kPa when the


319

800

Fibrous outer coat

Connective tissue

Force (N)

Adipose tissue

400

Figure 13.9

1.03 m s−1

200 0

Tuberosity

(a)

1.44 m s−1

600

Calcaneus

(b)

0

2

4 6 8 Displacement (mm)

10

(a) Section of a child’s heel and (b) force vs. displacement relation, for impact on a human heelpad (redrawn from Misevich and Cavanagh et al., 1984).

strain was increasing and ⫺30 kPa when the strain was decreasing. He concluded that the gas compression was close to isothermal, while the polymeric contribution was significant.

13.4.2 FEA of the heelpad and the foam midsole Figure 13.9a shows a section of the heel of a child. The calcaneus bone of the heel is cushioned by adipose tissue and by a fibrous outer layer. The total thickness averages 18 mm in adults. The fat is contained under pressure in a matrix of fibrous tissue. Misevich and Cavanagh et al. (1984) used a flat-faced pendulum, with a load cell on its face, to strike the heel at velocities of 1–2 m s⫺1. The force vs. displacement relation (Fig. 13.9b) is non-linear with hardening at high strains. The area inside the loading/unloading loop represents the considerable energy loss. Aerts et al. (1995) described the force–deflection response of the human heelpad, while Gefen et al. (2001) gave the mean pressure at the interface between the heelpad and a flat rigid surface as a function of an average heelpad thickness strain. However, they gave no heelpad stress–strain data. Verdejo and Mills (2004a) analysed the stress field under the heel, using ABAQUS static finite element analysis (FEA). In the axisymmetric geometry, with a vertical axis of rotational symmetry, the calcaneus bone was simplified to a hemisphere of radius 15 mm, attached to the end of a 20 mm long vertical cylinder of radius 15 mm. The heelpad is assumed to be bonded to the calcaneus surface, allowing some load to be transferred by shear to the cylindrical surface. The heelpad was simulated using the Ogden hyperelastic material, with shear moduli µ1 ⫽ µ2 ⫽ 50 kPa, exponents α1 ⫽ 30 and α2 ⫽ ⫺4, and inverse bulk moduli D ⫽ 1.0 ⫻ 10⫺7 Pa⫺1, which reproduced the

320 Polymer Foams Handbook Bone

Heelpad Axis of rotational symmetry

0.8 1.0

Steel cylinder 0.8 1.0 0.8 0.6 0.5

0.6 EVA foam

0.3 (a)

0

EVA 0 foam 0.4 0.2 Fixed steel table

Fixed rigid table 0.1 (b)

Figure 13.10 Contours of vertical compressive stress (MPa): (a) in the heelpad and EVA foam at a force of 500 N (Verdejo and Mills, 2004a) and (b) at the peak of an ASTM F1614 test on the same EVA foam midsole (Mills, unpublished).

heel-strike data of Aerts. The EVA midsole foam was taken as a vertical cylinder of radius 35 mm and height 22 mm, with flat end faces. The Ogden hyperfoam parameters µ1 ⫽ 1000 kPa, µ2 ⫽ 40 kPa, exponents α1 ⫽ 30 and α2 ⫽ ⫺4, and Poisson’s ratios ν1 ⫽ 0 and ν2 ⫽ 0.4 matched the tensile and compressive response of new EVA foam from a shoe midsole. Figure 13.10a shows the vertical compressive stress σ22 contours, at a total deflection of 10.2 mm, when the central foam deflection is 7.1 mm and the load is 0.50 kN. The heelpad has spread laterally across the concave upper surface of the midsole. The maximum foam stress of 700 kPa, at the centre of the foam upper surface, is about double that in the heel region of well worn trainers on a treadmill (Fig. 13.4a). A higher pressure is expected on the new midsole. The predicted force vs. deflection relationship (Fig. 13.11) is non-linear. In future, three-dimensional (3D) models of the foot will be used with viscoelastic material properties, to better simulate the heelpad and foam responses. Spears et al. (2005) considered a 3D bare heel during walking, and predicted realistic surface pressure distributions. However, the simple viscoelastic model had a single relaxation time. Yarnitzky et al. (2006) combined measurements of foot forces when subjects were walking on a treadmill, with two-dimensional (D) FEA of the heelpad deformation, to predict pressure distributions on the foot surface. They also pointed out that subjects varied considerably in heelpad thickness and elastic modulus.


321

1.2

1

ASTM test

Force (kN)

0.8 Foot and shoe model

0.6

0.4

0.2

0

0

5

10

15

Deflection (mm)

Figure 13.11 Predicted force vs. deflection of heelpad plus EVA midsole (Verdejo and Mills, 2004a) compared with dynamic FEA of the ASTM test on the same EVA (Mills, unpublished).

13.5

Durability Shoe test conditions must approximate those in running, with the stress levels, hence the same cell failure mechanisms. It is easy to apply a large stress to an EVA midsole and accelerated its deterioration. Figure 13.12 shows foam from the heel region of trainers after considerable use (Verdejo and Mills, 2004b). There are wrinkled cell faces and some small holes in some faces.

13.5.1 ASTM test for heel performance In the ASTM F1614 test, a runner’s heel is simulated by a vertical steel cylinder of 45 mm diameter; the flat end has a 1 mm radius edge. A total mass of 8.5 kg falls about 50 mm, imparting a 5 J impact energy. Dynamic FEA was performed of this test, using EVA material parameters described in Section 13.4.2, and a friction coefficient of 0.3 between the foam and the metal surfaces. Figure 13.10b shows that the test does not reproduce the stress distribution under a runner’s heel; the stress is too uniform in the central foam, with an unrealistically high peak of 2.0 MPa just below the cylinder radius. The latter is likely


Holes

(a)

(b)

Figure 13.12 The heel region of trainers after (a) 500 km showing some wrinkled cell faces and (b) 750 km with holes in some cell faces (Verdejo and Mills, 2004b).

to cause excessive damage to the foam in fatigue tests. The nearly linear predicted response is considerable stiffer than the deformable heel model, with a peak force of 1.1 kN (Fig. 13.11). Hence the ASTM test produces a higher peak force than a foot strike with the same kinetic energy.

13.5.2 Durability of EVA foams in repeated impacts When shoes are used for running, the foam pressure distribution is non-uniform, with maxima in the heel contact and metatarsal areas (Fig. 13.4a). The impacts, repeated at 0.3 s intervals, may cause fatigue damage to the foam. Misevich and Cavanagh (1984) repeatedly loaded EVA foam at a frequency of 1 Hz with an Instron. They cycled the load (on a flat-ended cylinder of diameter 44.5 mm) between 0 and 1780 N, and monitored the total input energy per cycle. The peak compressive stress was 1.15 MPa, which is significantly higher than typical in-use pressures. In real life, if the shoe foam response changes with time, and the person continued to run in the same way, the peak force would increase as a run proceeded, while the impact energy remained constant. Figure 13.13 shows how the peak input energy decreased linearly with the logarithm of the number of cycles in their experiment. The foam thickness at zero load became less as the test proceeded, and it is suspected that the cell gas content reduced. No comment was made about any temperature rise in the foam. Figure 13.13 could give the impression that the foam response change is permanent. However, EVA foam recovers slowly and partially after shoe use. Although scanning electron microscope (SEM) photographs of


323

Peak energy (J)

6

5

4

3 1

10

100 Compression cycles

1000

104

Figure 13.13 Peak input energy vs. cycle number for the compression of EVA foam midsole in heel of shoe at 1 Hz (redrawn from Misevich and Cavanagh, 1984 © Nike).

sections from old trainers show flattened cells near the foam lower surface, some of this is due to the initial processing; near-surface cells are flattened in the moulding process. Verdejo and Mills (2004b) performed experiments in which the foam was repeatedly impact loaded in compression at 1.5 Hz, while the stress and deflection were monitored. By plotting the loading part of the stress–strain curve to fit equation (11.4), the initial yield stress and the effective gas pressure of the foam were determined. Figure 13.14 shows that the former falls by about 30% over the simulated 1400 km of running, whereas the gas pressure remains almost constant. The peak stress increased from circa 460 to 540 kPa over the test duration. When athletes ran on the compliant surface of a treadmill (Verdejo and Mills, 2004a), the peak pressures increased both with the distance run, and with the time on the treadmill (Fig. 13.15) but were 30% lower than in the repeat impact test. The laboratory repeat-impact machine appears to reproduce the changes in the foam response observed in running. In contrast, Misevich and Cavanaugh’s use of a 1.2 MPa stress appears to be too severe.

13.5.3 Foam temperature rise in running The foams have temperature sensitive properties, but this is often ignored. Kinoshita and Bates (1996) measured the temperature rise


Initial yield stress (kPa)

120

100

80

60

40 0

500

1000 Run distance (km)

1500

0

500 1000 Run distance (km)

1500

(a)

Gas pressure (kPa)

300

250

200

150

100 (b)

Figure 13.14 Variation of the initial yield stress and effective cell gas pressure with simulated run distance (Verdejo and Mills, 2004b).

in midsole foam, for runners in Japan under a variety of conditions. There is heat transfer from the tarmac road surface and from the foot. Typically the foam temperature rose to a plateau value within 20 min of the onset of running (Fig. 13.16). The foam also heats by hysteresis, since the maximum midsole temperature exceeded the higher of the exterior air temperature and the road temperature by 5–10°C. Since the EVA foam has a melting temperature Tm of about 70°C, its mechanical properties change as Tm is approached. It seems that heat generation and heat flow should be part of a complete model of shoe foam performance. Kinoshita and Bates suggest that athletes select shoes with the appropriate EVA foam hardness for summer or winter use.


325

350 325

Tekscan pressure (kPa)

300 275 250 225 200

1 min running 5 min running

175

10 min running

150

0

100

200

300

400

500

Run distance (km)

Figure 13.15 Peak pressure in a runner’s heel region as a function of the run distance and the time on a treadmill (Verdejo and Mills, 2004a).

60

Temperature (°C)

50

40

30

20

10

0

0

10

20 Time (min)

30

40

Figure 13.16 EVA midsole temperature rise for seven cases of running in Japan (Kinoshita and Bates, 1996).


13.6

Discussion The analysis of foam performance in running shoes has progressed both with the use of in-shoe pressure mapping and with FEA analysis of the stresses in the shoe. Further work is necessary on the heel tissue mechanics. The selection of EVA foam is a compromise. It is a low cost material, while its low Young’s modulus allows a wedgeshaped midsole to bend relatively easily, while cushioning the impact of various parts of the foot. However, its durability to repeated impacts is not ideal. Designs, where separate components cushion the heel strike, are more costly and it is not obvious that they improve the biomechanics of running.

References Aerts P., Ker R.F. et al. (1995) The mechanical properties of the human heel pad, J. Biomech. 28, 1299–1304. Alley L. & Nichols G. (1999) Future technical challenges for PU industry in athletic footwear production, Utech Asia conference, Singapore. Barlett R. (1995) Sports Biomechanics, Spon. Bobbert M.F., Yeadon M.R. & Nigg B.M. (1992) Mechanical analysis of the landing phase in heel–toe running, J. Biomech. 25, 223–234. Diegritz W. (1998) Well padded, Kunststoffe-German Plast. 88, 1494–1496. Frederick E.C. (1984) Sports Shoes and Running Surfaces, Human Kinetics Publ. Inc., Champaign, IL. Gefen A., Megido-Ravid M. et al. (2001) In vivo biomechanical behavior of the human heel pad during the stance phase of gait, J. Biomech. 34, 1661–1665. Hennig E.M. & Milani T.L. (1995) In shoe pressure distribution for running in various types of footwear, J. Appl. Biomech. 11, 299–310. Hunter S., Dolan M.G. et al. (1995) Foot Orthotics in Therapy and Sport, Human Kinetics Publ. Inc., Champaign, IL. Indesteege J., Camargo R.E. et al. (1997) Considerations on the selection of PU systems for midsole wedges in athletic footwear applications, Polyurethanes World Congress. Amsterdam. Kinoshita H. & Bates B.T. (1996) The effect of environmental temperature on the properties of running shoes, J. Appl. Biomech. 12, 258–268. Mills N.J. (2003) Running shoe materials, Chapter 4, in Sport Materials, Ed. Jenkins M.J., Woodhead, Cambridge, pp. 65–99. Misevich K.W. & Cavanagh P.R. (1984) Material aspects of modeling shoe/foot interaction, in Sports Shoes and Playing Surfaces, Ed. Frederick E.C., Human Kinetics Publ. Inc., Champaign, IL.


327

Nigg B.M., Ed. (1986) Biomechanics of Running Shoes, Human Kinetics Publ. Inc., Champaign, IL. Pillow J.G. (1997) Ethylene elastomers made using constrained geometry catalyst technology, IRC 97 Conference, Nuremburg, Germany. Robbins S. & Waked E. (1997a) Hazard of deceptive advertising of athletic footwear, Br. J. Sport. Med. 31, 299–303. Robbins S. & Waked E. (1997b) Balance and vertical impact in sports: role of shoe sole materials, Arch. Phys. Med. Rehabil. 78, 463–467. Sawai M., Miyamoto K. et al. (2000) Super low density polyurethane systems for sports shoes, J. Cell. Plast. 36, 286–293. Segesser B. & Pforringer W., Eds. (1989) The Shoe in Sport, Year Book Medical Publ., Chicago, IL. Shorten M.R. (1993) The energetics of running and running shoes, J. Biomech. 26(Suppl. 1), 41–51. Shorten M.R. (2000) Running shoe design: protection and performance, in Marathon Medicine, Ed. Tunstall Pedoe D., Royal Society of Medicine, London, pp. 159–169. Sims G.L.A. & Khunniteekool C. (1996) Compression moulded ethylene homo- and copolymer foams, J. Cell. Polym. 15, 1–14, 15–29. Spears I.R., Miller-Young J.E. et al. (2005) The effect of loading conditions on stress in the barefooted heel pad. Med. Sci. Sport. Exerc. 37, 1030–1036. Swigart J.F., Erdman A.G. & Cain P.J. (1993) An energy based method for testing cushioning durability of running shoes, J. Appl. Biomech. 9, 26–47. Torden M.J. (1993) Evaluation of the cushioning of closed-cell plastic foams, Dynamic Loading in Manufacturing and Service, Inst. Eng. Aust. Conf., Institution of Engineers, Australia, Melbourne, 11 pp. Verdejo R. & Mills N.J. (2002) Performance of EVA foams in running shoes, 4th International Conference, The Engineering of Sport, Kyoto, 7 pp. Verdejo R. & Mills N.J. (2004a) Heel–shoe interactions and the durability of EVA foam running shoe midsoles, J. Biomech. 37, 1379–1386. Verdejo R. & Mills N.J. (2004b) Simulating the effect of long distance running on shoe midsole foam, Polym. Test. 23, 567–574. Yarnitzky G., Yizhar Z. & Gefen A. (2006) Real-time subject specific monitoring of internal deformations and stresses in the soft tissue of the foot, J. Biomech. (in press).

Chapter 14

Bicycle helmet case study

Chapter contents 14.1 Introduction 14.2 Biomechanics criteria for head injuries 14.3 Bicycle helmet standards 14.4 Materials and process selection 14.5 Thermal comfort and fit 14.6 Design of helmets for impacts 14.7 Bicycle helmet effectiveness 14.8 The future References

330 331 333 335 337 339 348 348 349


14.1

Introduction Cycling, a healthy method of transport, involves higher speeds than walking. In many countries cyclists share the road with cars and other vehicles, with a consequent risk of collision. Head injuries are more likely to be life threatening than broken limbs, and the medical profession can rarely reverse the effects of brain damage. The use of bicycle helmets reduces the number and severity of head injuries and deaths in crashes (the term crash is preferred to accident, which suggests that no one is to blame). One strategy for reducing injuries is to persuade cyclists to wear helmets. In the UK, surveys (Gregory et al., 2003) show that the wearing rate is about 25%. Countries such as Australia and New Zealand have legislated for compulsory bicycle helmets. Helmets cannot prevent all head injuries; the aim is to minimise the social costs (of medical treatment, long-term care, loss of employment, etc.) of injuries to this group of road users. Other strategies include modification of cyclists’ behaviour by training or enforcement of the traffic laws. In countries where cyclists are segregated from motorists, bicycle helmets are less necessary. This case study considers materials selection, and the design of helmets under the constraints of mass, size, and cost. Figure 14.1 Comfort foam

Ventilation hole

Microshell

EPS liner Headband

Figure 14.1

Retention straps

Components of a bicycle helmet, seen in cross-section. A ratchet mechanism at the rear of the headband adjusts the fit of the single liner size to the range of head sizes. EPS: expanded polystyrene.

Chapter 14 Bicycle helmet case study

331

shows the components of a typical bicycle helmet; we will concentrate on the foam liner. Most helmets have a thin thermoplastic shell (a microshell) over the foam moulding.

14.2

Biomechanics criteria for head injuries Serious brain injuries can be attributed to three main causes: 1. Skull fractures: Most are linear fractures, caused by an impact on a rigid flat surface; the bending stresses in the skull exceed its tensile strength. Depressed fractures are caused by a rigid convex object penetrating the skull. Helmets should prevent fractures by distributing and attenuating the force applied to the head. Experiments with an egg show that, if the load is distributed (using expanded polystyrene (EPS) foam on both sides), the compressive failure load is very high, whereas if it is localised, the failure load is low. Bone fragments from skull fractures can damage the exterior surface of the brain, with a risk of causing epilepsy. 2. Linear acceleration of the brain: Direct blows to the skull cause the brain, with the modulus of a soft jelly and floating in cerebrospinal fluid, to move and distort. When the surfaces of the brain, particularly the temporal and frontal lobes, impact the interior of the skull, they are prone to bruising. Blows to the side of the head appear to cause more severe brain injuries than frontal blows with the same acceleration levels. 3. Rotational acceleration of the brain: High levels of rotational acceleration applied to the heads of cats and baboons caused concussion, or permanent brain damage of a diffuse nature (Bandak et al., 1996). The animal’s head was held in a cage, the rotational acceleration of which was ramped to a constant level; consequently no linear acceleration was applied. Size scaling suggests that the rotational acceleration, to cause concussion in humans, is the order of 10,000 rad s⫺2. In real head impacts, the rotational acceleration can vary rapidly with time, and the peak values alone may not determine the severity of head injury. The net impact force in a crash rarely acts through the centre of gravity of the head, so it causes both linear and rotational head acceleration. For more detail of the injury mechanisms, and how the brain motions can be modelling see (Kleiven, 2002). In the initial stages of a frontal or lateral head impact, the head moves in a straight line without rotation, while the neck ‘shears’; only later does the neck reach its limit of flexure, causing the head to rotate. The injury-causing head accelerations occur in the initial phase of the impact. During this, the neck reaction force is much smaller than the impact force on the head, so the torso mass has little effect on the peak


Force (kN)

No helmet

With helmet Time (ms)

Figure 14.2

Schematic of the force time trace, for a vertical fall to the road, with and without a helmet.The area under the two graphs is the same.

head acceleration. Gilchrist and Mills (1996) showed, using a dummy with a realistically flexible neck, the peak head acceleration was equal to the peak head contact force divided by its mass. Therefore it is reasonable to test helmets using a free headform rather than with a full dummy. To protect the head against injury, the helmet should: (a) Resist complete penetration by convex objects. The local pressure on the skull should not exceed circa 2.5 MPa, or there could be local skull fracture(s). (b) Prevent the impact force component, acting towards the centre of gravity of the head, exceeding 10 kN. This limits the peak linear acceleration of a 5 kg head to less than 200g. (c) Prevent the impact force component, tangential to the helmet surface, exceeding 2 kN. This will keep the angular head acceleration below the estimated limit for injury. Helmets are designed mainly to meet the linear acceleration condition. Let us consider a vertical fall, with and without a helmet. By Newtonian mechanics, the impulse (the area under the graph in Fig. 14.2) required to reduce the head’s momentum to zero when it strikes the road is the same. With a helmet, it is spread over a longer time, due to the crushing of the foam liner; consequently the peak force is lower. The designs tend to easily meet the skull fracture condition. In oblique impacts with the road surface, sliding at the helmet/road and helmet/hair interfaces, and helmet liner crushing reduce the magnitude of the peak tangential force on the head surface (see Section 14.6.4). Consequently, the rotational acceleration of the head is less with a helmet than without one. Forensic reconstructions of bicycle helmet damage by Smith et al. (1994) and McIntosh et al. (1996) indicate that the acceleration levels, to cause short-term concussion, range from 100 to 200g. They show that the majority of bicycle helmet damage occurs for impact


333

AS 2063.2 limit (v = 5.42 m s−1)

400

Peak headform acceleration (g)

350 −1)

ANSI Z90.4 limit (v = 4.57 m s−1) Snell B90 Limit (v = 6.32 m s

300

CSA D113.2 limit (v = 5.70 m s−1)

250 200 150 100 50 0 0

Figure 14.3

1

2

3 4 5 Impact velocity (m s−1)

7

Estimated head accelerations from helmet damage reproduction, and test criteria of various standards. (Smith et al., 1994 © Elsevier). Helmeted

Figure 14.4

6

Non-helmeted

Head impact sites on bicyclists in Sweden (Larsen et al., 1991 © IRCOBI).

velocity less than 5.4 ms⫺1, equivalent to free falls from 1.5 m or less on to a rigid flat object (Fig. 14.3). Surveys show that the road surface is commonly impacted, and that the most common impact sites are on the front and sides of the head (Fig. 14.4).

14.3

Bicycle helmet standards The jargon, used in standards, can be difficult for the lay reader to understand, and the test equipment may be difficult to visualise from the outline drawings provided. Test methods with a low capital cost must produce consistent results. There are direct impacts with a rigid flat anvil and a kerbstone-shaped anvil (with 105° included angle and

334 Polymer Foams Handbook 15mm radius edge). Although some cyclists impact deformable steel panels on vehicles, and vehicle glazing, there are no tests on deformable anvils. Table 14.1 compares several national standards, as does the web site www.bhsi.org/webdocs/stdcomp. The drop height, which determines the kinetic energy of the impact, should relate to the head height while riding of about 1.5 m, however it varies considerably. Only the EN 1078 standard, of those in Table 14.1, uses a headform that is free to rotate; the others use a headform clamped to a neck which cannot rotate. As with motorcycle helmet impacts (Section 16.3) and corner impacts on packages (Chapter 12), when the dropped object is free to rotate, the peak linear acceleration is somewhat smaller than when it is constrained to the same angular position. Hence the allowed peak acceleration of 250g in EN 1078 is roughly equivalent to the 300g for the standards that use a non-rotating headform. The human head mass increases with circumference, as do the headforms in the EN, but not the US Consumer Product Safety Commission (1998) or Snell (1995)

Table 14.1 Requirements for bicycle helmet impacts Standard

Snell B95

ASTM

CPSC

EN 1078

Drop (m) onto flat anvil Drop (m) onto kerbstone (K) or hemispherical (H) anvil Permissible peak acceleration (g)

2.2* 1.44* (H)

2.0 1.2 (H), 1.2 (K)

2.0 1.2 (H), 1.2 (K)

1.5 1.06 (K)

300

300

300

250

* The impact energy is specified; this is the drop height if the headform mass ⫽ 5 kg.

10° AA plane Reference plane

Headform

Figure 14.5

Impact test sites in EN 1078 are in the shaded area (redrawn from EN 1078).


335

standards. Hence it is difficult to compare the protection levels of these standards. The tests are minimum requirements; comparative test results are occasionally published in consumer magazines. The need to meet impact test criteria at 50°C and at ⫺20°C, with different anvils, means that the foam often limits head accelerations to the range 150–200g at 20°C. The impact sites in EN 1078 (Fig. 14.5) are not as low at the front and sides as in real crashes (Fig. 14.4). The horizontal reference plane is meant to pass through the notch in the ear below the ear opening and the lower edges of the eye sockets. However, for people of the same head circumference, the position of this plane is a variable vertical distance from the crown (Peoplesize Software). The AA plane is 25 mm above the reference plane.

14.4

Materials and process selection 14.4.1 Foam liner Many low-density materials can absorb significant amounts of energy in compression without the stress exceeding 2.5 MPa (the local pressure that might cause skull fracture). The choice of materials is based on this parameter, the cost, ease of processing, and minimising the helmet weight. Cork, used in pre-1960 motorcycle helmets, is only available in limited sizes, is expensive, and is denser than polymer foams of the same compressive collapse stress. Flat sheets of honeycomb, made from polymer, paper, or aluminium foil, are optimal for energy absorption when compressed along the cell axes. However, the sheet cannot be bent to a doubly convex shape of a helmet. Honeycomb also needs a protective outer shell to avoid being split parallel to the cell axes by a wedge-shaped object. Anisotropic cellular materials, with optimal energy absorption direction in one direction, tend only to exist as flat sheets. The yield stress of closed-cell foams varies approximately with the 1.5th power of the foam density (Fig. 14.6), and is proportional to the polymer yield stress. Glassy polystyrene (PS) has a higher compressive yield stress (circa 100 MPa) than rigid polyurethane (PU), so is preferable. Holes can be moulded in EPS liners, whereas if crosslinked polyolefin foams were thermoformed, the holes would need to be cut in a post-moulding operation. Therefore most bicycle helmets are moulded from EPS beads, with a small minority using expanded polypropylene (EPP) or PU foam castings. The moulds for casting rigid PU foam have a low capital cost, but the process is slow, and the liner density is higher than that of EPS of the same yield stress. Although EPP liners cost several times as much to manufacture as EPS, the foam is

Compressive yield stress (MPa)


1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

0.2 20

Figure 14.6

Slope 1.78 30

40 50 60 Foam density (kg m−3)

70

80 90 100

Variation of compressive yield stress of EPS with density (Mills, unpublished).

less brittle and recovers better after an impact. It may be preferred for skate-board helmets, which suffer a large number of minor impacts. EPS only recovers a little after impact, so the helmets should be destroyed after a crash. EPS bicycle helmets are imported for less than $10 (£5). The mass of typical bicycle helmets is 0.25–0.35 kg; any extra mass adds to discomfort.

14.4.2 Microshell The outer shell prevents foam damage in everyday use, and foam fracture in a crash. In contrast with motorcycle helmets, it plays a minor part in energy dissipation. Some circa 1990 designs had either a thin cloth cover or no cover over the EPS liner. Some ‘soft-shell’ bicycle helmets broke up in crashes (McIntosh and Dowdell, 1992), while others broke up in BS impact tests, particularly after water immersion. Bending of the EPS liner often initiates cracks at the sides of the crushed area. If the impact is on one of the ventilation holes, the effective crack length is half of ventilation hole length plus the surface (bead boundary) crack length. The wedging force needed to propagate this crack is of the order of 50 N. As the crushing force in a bicycle helmet impact is many kN, foam fractures are possible. The fracture toughness of EPS is low (Chapter 15), so cracks can propagate and split no-shell helmets into several pieces. A thermoplastic microshell provides tensile strength to the helmet. It is often a polycarbonate thermoforming 0.5 mm thick, which can


(a)

337

(b)

Figure 14.7

(a) Microshell interior, with transferred parts of EPS beads and (b) sectioned liner, showing an insert to mount a chin strap end in the EPS (Mills, unpublished).

be in-mould bonded to the EPS, or its edges attached to the foam by adhesive tape. It makes the indentation of the foam more difficult in oblique impacts, reducing the force tangential to the helmet surface. Figure 14.7a shows the interior of a bonded microshell that has been pulled from a liner; the failure occurred through the EPS beads, rather than at the EPS/shell interface. Finite element analysis (FEA) Section (14.6.3) shows that a microshell does not absorb much energy in a crash, however it affects the deformation of the liner and prevents fracture. Some helmets have thermoplastic frameworks moulded into the EPS for reinforcement. Figure 14.7b shows an insert in the EPS that anchors the end of a chin strap.

14.5

Thermal comfort and fit 14.5.1 Ventilation About 30% of the body’s heat loss is from the head; as foam is a good thermal insulator, air-flow ventilation is necessary for comfort in hot weather. For effective ventilation, the holes at the front of the helmet must be large, and there must be fore and aft channels in the interior surface to allow air flow past the hair and scalp. Brühwiler et al. (2006)

338 Polymer Foams Handbook evaluated the power dissipated from a headform in a wind tunnel. The headform was heated and could perspire. They showed that cooling of the face and forehead was important, so vents should cause an air flow over the forehead. Helmets varied in their heat transfer efficiency. Current helmets have far more and larger ventilation slots than helmets from the mid-1990s. They often have extensions at the rear for streamlining, and sometimes a non-smooth external profile (Fig. 14.8).

14.5.2 Fit and retention The helmet must stay in place in a crash, in spite of the top of the head acting like a ball and socket joint with the helmet interior. The microshell is insufficiently strong for the chin straps to be directly attached. Therefore the 20 mm wide polyester webbing straps are often passed through slots moulded in the foam (Fig. 14.1), allowing high crash forces (1 kN) to be taken by the foam in compression. Straps run to the front and rear of the helmet; the front straps tighten if the helmet rotates rearward, and the rear straps tighten if the helmet rotates forward. This is only effective if the strap is tightened securely under the chin, which may be slightly uncomfortable. If the helmet is pushed back by the wearer to expose their forehead, this could drastically reduce the head protection at frontal sites in a crash.

Figure 14.8

Rear view of ventilation holes in specialised S1 helmet from 2004.


339

Figure 14.8 shows a hinged thermoplastic moulding, that grips the head at the base of the occiput, hence anchors the helmet in place. Many helmets have adjustable headbands that have the same role. The microshell reinforces areas where chin straps pass through or round the EPS liner; a chin strap force of 1 kN in a crash might cause an unreinforced liner to fracture. Although the retention system allows some helmet rotation on the head, this is beneficial as it reduces the head rotational acceleration.

14.6

Design of helmets for impacts The design of bicycle helmets is simpler than that of motorcycle helmets, since the microshell causes little load spreading and its low mass does not give rise to force oscillations. Although neither the human skull nor the outer surface of a helmet is exactly spherical, it is a reasonable approximation that the region where the impact occurs is spherical. Both the skull and the road surface are treated as being rigid, while the helmet shell is assumed to have negligible bending stiffness. The 5–8 mm thick scalp, which is soft and deformable, is also ignored in the design. Four cases are considered. The first is an over-simplification, but readily leads to a design. The second, in which FEA is used on helmets without ventilation holes, shows the principles of load spreading. The last two cases use dynamic FEA on a full helmet shape, and provide the justification for the design.

14.6.1 Approximate model for direct impacts The contact geometry between a flat rigid surface (the road) and a head of radius R is shown in Figure 14.9. There are no ventilation

Head

Rotational symmetry axis R−x

R

a x

Figure 14.9

Uncrushed foam zero stress Crushed foam uniform high stress

Rigid flat surface

Geometry of head, and helmet foam crushing, showing assumed stress contours in the foam.

340 Polymer Foams Handbook holes in the helmet (typical in circa 1990). The liner crush distance x is much less than R (100–200 mm), since the liner thickness less than 30 mm. The foam crushes over a disc of radius a. Applying Pythagoras’s theorem to the triangle gives R2 ⫽ (R ⫺ x)2 ⫹ a2 If the x2 term is ignored in the expansion of the brackets, the contact area A is A ⫽ π a2 ⫽ 2πRx

(14.1)

Assuming no load spreading (Fig. 14.9), and that the foam has a constant yield stress σy while the strain is increasing, Mills (1990) deduced that the force F transmitted by the foam is F ⫽ Aσy ⫽ 2πRσy x

(14.2)

The slope of this straight line is the loading slope k. Substituting typical values of R ⫽ 140 mm for the front of a helmet liner, and σy ⫽ 0.7 MPa gives k 艑 600 N mm⫺1. For an average-size headform of mass 5 kg, undergoing a 2 m drop, the kinetic energy at impact is E ⫽ 100 J. Although EN 1078 allows a 250g headform deceleration, a margin must be allowed for material variability and tests at high and low temperatures. Hence the design acceleration limit is 200g at 20°C. This is equivalent to the condition that F ⬍ 10 kN (Fig. 14.10). The other limit in the figure is for the foam bottoming out. This occurs when the compressive strain approaches 1 ⫺ R (R is the foam relative density), and the stress rises rapidly. Typically bottoming out occurs at a strain of 80%. The foam thickness T must be a minimum of 1.25xmax to avoid this. The area under the force–deflection curve is equal to the energy input. If the force reaches 10 kN, as the headform comes to a momentary halt at deflection xmax, and equation (14.2) applies, then 0.5 ⫻ 10 kN ⫻ xmax mm ⫽ E

(14.3)

Therefore the optimum foam is one where the loading line meets the intersection of the two ‘dangerous’ areas. If lower yield stress foams are used, they will bottom out before the load reaches 10 kN; if the foam has too high a yield stress, the force will reach 10 kN before the foam bottoms out.


341

Concussion likely

Force on head (F )

10 kN Too high

Optimal

Foam bottoms out

Too low Foam deflection (x)

0.8 T

Figure 14.10 Responses for a bicycle helmet hitting the road surface, for three values of the foam yield stress.

The sequence of design is that: (a) Given the impact energy E, equation (14.3) is used to find xmax. The helmet mass (⬍0.3 kg) is ignored. For E ⫽ 100 J, xmax ⫽ 20 mm. Allowing for bottoming out, the foam thickness T ⫽ 25 mm. (b) Using the helmet radius at the impact site and equation (14.2), the foam yield stress is calculated to give a loading curve that passes through the point (80% of foam thickness, 10 kN) (Fig. 14.10). (c) The polymer and the foam density are chosen to provide the appropriate yield stress. For an impact site of radius 100 mm, the yield stress required is 0.7 MPa, hence the density of EPS should be 65 kg m⫺3 (Fig. 14.6). (d) The loading curves are recalculated for the extremes of the test temperature range in the standard, using data for the foam at ⫺20°C and 50°C. For a design impact velocity V, the foam thickness should be proportional to V2, and the foam yield stress to V⫺2, to keep the head acceleration below 200g.

14.6.2 FEA of direct impacts on unventilated helmets Mills and Gilchrist (2003) used static FEA on a simplified helmet with a uniform 30 mm thickness PS foam liner, of density 35 kg m⫺3. The headform, with 120 mm radius of curvature in the fore-and-aft

342 Polymer Foams Handbook Kerb stone

0.4 Flat anvil

0.3 0.1

0 PS foam

0.4 0.2 0.6

0 PS foam

Mirror plane

Mirror plane

Inner surface Inner surface

(a)

(b)

Figure 14.11 Contours of the vertical compressive stress in MPa: (a) flat surface impact and (b) kerbstone impact (seen end on) at sites 30° from the helmet crown. Only the rear half of the helmet liner is shown (Mills and Gilchrist, 2003).

direction, and 80 mm in the side-to-side direction, exactly fitted the helmet liner interior. The impact site was off-centre, in the coronal plane (the vertical plane containing both ears). Impacts were with a flat rigid plane, the kerbstone of EN 1078, or a rigid hemisphere of 50 mm radius. For a flat surface impact (Fig. 14.11a), the compressive stress contours are nearly parallel to the force vector, on the mirror symmetry plane, through the centre of the impact site. The stress is nearly constant in the contact area, and decreases rapidly at the sides, confirming the approximation of Figure 14.9. For the kerbstone anvil (Fig. 14.11b), the foam stress increases towards the centre of the contact area, with non-zero stresses well outside the contact area, so the stress distribution differs from that in Figure 14.9. Figure 14.12 compares the loading curves for three anvils. The impact sites are 30° from the crown for the kerbstone and flat anvils, but 60° for the hemisphere (close to the helmet lower edge). The responses are nearly linear, confirming the simple relationship of equation (14.2). The loading slopes are 337 (flat), 150 (kerbstone), and 81 N mm⫺1 (hemispherical anvil). Equation (14.2), which predicts a slope of 251 N mm⫺1 for a mean head radius of 100 mm and an initial yield stress of 0.4 MPa, underestimates the slope for the flat anvil by 34%. As a rule of thumb, the loading curve slope for a kerbstone impact is 50% of that for a flat surface impact, but this depends on the orientation of the kerbstone relative to the helmet. A foam, that is ideal for an impact with a flat surface at a given velocity, has a suboptimal yield stress for an impact on a kerbstone at the same velocity. At the side of the helmet, where the radius of curvature is larger, the foam geometry may need to be altered. One method is to incorporate ventilation slots, thereby reducing the contact area between the foam


343

4 3.5

Kerb

Flat

Force (kN)

3 2.5 2 1.5 1

Hemi

0.5 0

0

5

10

15

20

25

30

Displacement (mm)

Figure 14.12 Predicted force vs. foam crush distance for a bicycle helmet using XPS 35 foam, on three anvils (Mills and Gilchrist, 2003). 5 Flat on front Flat on side

Striker force (kN)

4

3

2

Kerb on side

1

0

0

10

20

30

Helmet deflection (mm)

Figure 14.13 Force–deflection curves for a bicycle helmet impacted at the side and front onto flat and kerbstone anvils (Mills, 1990).

and the head. However, many helmet designs have large vents in all parts of the helmet, so alternative solutions may be used. Mills and Gilchrist (1991) impacted the front and side of bicycle helmets (with no ventilation holes in the impact region) on a rigid flat surface. One helmet, containing 68 kg m⫺3 density PS foam with a measured yield stress of 1.06 MPa, had a 2 mm thick high-density polyethylene (HDPE) shell. Its loading response is linear (Fig. 14.13), in agreement with the simple theory of equation (14.2). The theoretical


Loading curve slopes for impacts on a bicycle helmet N mm⫺1

Impact conditions Side on flat anvil Side on 50 mm radius hemisphere

1.6

3

711 350

1.2 0.8 0.4

2 3

2

1

65 1

(a)

50 35

20

(b)

Figure 14.14 FEA of impacts at VV ⫽ 5.4 onto a flat anvil, with stress contours (MPa): (a) compressive σ22 on the liner outer surface and (b) tensile principal stress in the microshell (Mills and Gilchrist, 2006a).

slope using equation (14.2) was 59% higher than the experimental value for the impact on a flat surface. The loading curve slope, for a frontal impact on a flat surface, is lower (309 N mm⫺1) than for a side impact, reflecting the smaller radius of curvature of the helmet at the front (Table 14.2).

14.6.3 FEA of direct impacts on ventilated helmets To realistically simulate bicycle helmet performance, the exact shapes of the helmet and headform must be considered, as well as details of the retention system. Mills and Gilchrist (2006a) used the scanned shapes of a specialised S1 helmet, the rear view of which is shown in Figure 14.8, and a headform. A deformable headform was initially considered, but vibrations in it made the mean acceleration difficult to determine. As the headform deformation is small, a rigid headform, with mass and angular inertia equal to that tested, was simulated. The polyester webbing chin-straps were tensioned in a separate FEA, to accurately follow the facial shape of the headform. Frictional conditions at the head/helmet and helmet/road surface were considered. Figure 14.14a shows the compressive stresses are spread over a large region of the helmet liner, at the peak of a direct impact as in


345

x z FN

y 3

1 2

L1

Figure 14.15 As the impact force FN in a lateral impact does not go through the headform centre of gravity (origin of xyz axes), it causes rotational as well as linear acceleration (Mills and Gilchrist, 2006c).

EN 1078; the pattern is complex due to the ventilation holes. The tensile principal stress in the microshell (Fig. 14.14b) is close to the yield stress in the impact region, showing that the microshell restrains the spreading of the EPS. Without a microshell, the EPS was predicted to spread laterally in the impact region, allowing the headform to penetrate towards the road. In general, the forces on the helmet do not pass through the centre of gravity of the headform (Fig. 14.15), so there is rotational acceleration of the headform. The impact force from the road is a linear function of the reduction in (the closest) distance between the headform and the road, confirming the prediction of the simple helmet models.

14.6.4 FEA of oblique impacts on ventilated helmets Mills and Gilchrist (2006a) performed FEA of a bicycle helmet making an oblique impact with the road surface. This confirmed the implicit assumption of standards, that the horizontal component of the impact velocity has little effect on the peak linear head acceleration; the latter was largely determined by the vertical velocity component. Oblique impacts can be performed on sites observed in crashes (Fig. 14.16), such as the forehead, without damaging the test equipment. If a cyclist falls forward, striking the front of his helmet on the road, the frictional forces on the headform act to prevent the face rotating towards the road. The magnitude of the peak head rotational accelerations only increased slightly with the horizontal component of the


(a)

(b)

Figure 14.16 Predicted helmet rotation for an oblique impact (Mills and Gilchrist, 2006c).

impact velocity. Even for vertical falls, there is a significant peak rotational acceleration, due to the offset of the head centre of gravity from the line of action of the impact force.

14.6.5 Experimental oblique impacts on ventilated helmets In oblique impact tests (Mills and Gilchrist, 2006b), the correlation between the normal and tangential forces on an instrumented ‘road’ surface (Fig. 14.17) suggests an effective friction coefficient of about 0.2. High-speed film shows that there is sliding followed by rolling of the helmet shell on the road, plus sliding at the helmet/headform interface. FEA, which takes account of the shear and compression of the plasticised PVC headform scalp, confirms the sliding at the two interfaces. The foam liner, by crushing and thereby reducing the peak normal force (Fig. 14.2), consequently reduces the peak tangential force at the helmet/road interface. Therefore, wearing a helmet reduces the level of rotational head acceleration in an oblique impact with the road surface. When the headform force was plotted against the helmet deformation (estimated by integrating the headform acceleration), for an oblique impact, there is still a linear loading portion of the graph (Fig. 14.18). Therefore, in spite of the complex pattern of ventilation holes, and the oblique nature of the impact, the response is similar to that shown in Figure 14.13 for a direct impact on a helmet without ventilation holes.


347

1.4

Tangential force FT (kN)

1.2 1 0.8 0.6 0.4 0.2 0

0

1

2 3 4 Normal force FN (kN)

5

6

Figure 14.17 Variation of tangential with normal force at the shell/road interface for oblique impact of Arc helmet on the frontal 90° site (Mills and Gilchrist, 2006b).

5

Headform force (kN)

4

3

2

1

0

0

5

10

15

20

25

Helmet deformation (mm)

Figure 14.18 Headform impact force vs. liner deformation for right 70° oblique impact with VH ⫽ 3.6 m s⫺1 and VV ⫽ 4.5 m s⫺1 on flat surface, for an Aventicum helmet (Mills and Gilchrist, 2006b).


14.7

Bicycle helmet effectiveness The effectiveness of bicycle helmets has been assessed by epidemiological studies. Thompson et al.’s (1999) review of worldwide research concludes ‘Helmets provide a 63–88% reduction in the risk of head, brain, and severe brain injury for all ages of bicyclists. Helmets provide equal levels of protection for crashes involving motor vehicles (69%) and crashes from all other causes (68%)’. The Snell Foundation in the USA partly commissioned the Harborview survey of 527 helmets (Rivara et al., 1996). They comment ‘Brain injury increased very slightly with increasing (helmet) damage score up to the point where the helmet received catastrophic damage. Then the injury rates shot up dramatically.’ This means that any impact, which does not cause the bicycle helmet foam to completely crush, will only cause a minor head injury. The web site www.bhsi.org lists much of the research literature, both in favour and against the use of bicycle helmets, together with medical publications on cyclists’ head injuries. Hillman (1993) argued that a cyclist, wearing a helmet, may take greater risks – increasing his speed until his perceived risk is the same as before. He argues that bicycle helmets cannot be relied upon to protect riders from all the types of head injury: ‘To avoid impairing vision or hearing, bicycle helmets are designed to be worn high on the head and thus do not afford protection to parts of the head, neck, and upper face which account for half of the so-called “head” injuries of cyclists.’ He also argued, without evidence, that they do not protect the head from rotational trauma. Spaite et al. (1991) showed that wearing a helmet reduces the rate of injury to parts of the body other than the head; the inference is that helmet wearers are careful riders. Australia, New Zealand, and several states in the USA have made the wearing of bicycle helmets compulsory. The change in Australia in July 1990 caused a significant reduction in head injuries, but some can be attributed to a reduction in the number of cyclists on the roads. Recent New Zealand statistics show a drop in pedestrian injuries in parallel with the drop in cyclists’ injuries, in spite of the former group not wearing helmets. Statistical and risk compensation arguments against helmet legislation were challenged by Hagel et al. (2006).

14.8

The future Bicycle helmets could provide greater levels of protection, and/or protect more of the head (Mills and Gilchrist, 2006c). A helmet of thickness 50 mm could provide head protection for a fall from 4 m onto a road surface, or greater speed impacts with vehicles.


349

However, it might be unacceptable ergonomically or aesthetically. The mass would be only 25% greater, since an EPS of 63% of the density would have 50% of the compressive yield stress as the EPS used in the current 25 mm thick helmets. There would be greater thermal insulation. Some high-speed impacts, involving contact with vehicles, will be outside the protective capacity of the helmets. Helmet foam optimisation depends on the impact velocity and the nature of the object struck, so it is not possible to have an optimum design for all impacts.

References Bandak F.A., Eppinger R.H. & Omaya A.K., Eds. (1996) Traumatic Head Injury, Mary Ann Liebert Inc., Larchmont, NY. Brühwiler P.A., Buyan M. et al. (2006) Heat transfer variations of bicycle helmets, J. Sports Sci. 24, 999–1011. Consumer Product Safety Commission (1998) 16 CRF Part 1203, Safety standard for bicycle helmets, Fed. Regis. 63, 11712– 11747. EN 1078 (1997) Protective Helmets for Pedal Cyclists, British Standards Institution, London. Gilchrist A. & Mills N.J. (1996) Protection of the side of the head, Accid. Anal. Prev. 28, 525–535. Gregory K., Inwood C. & Sexton B. (2003) Cycle helmet wearing in 2002. TRL Report 578, Crowthorne, Surrey. Hagel B., Macpherson A. et al. (2006) Arguments against helmet legislation are flawed, BMJ 332, 725–726. Hillman M. (1993) Cycle Helmets: The Case For and Against, Centre for Policy Studies, London. Kleiven S. (2002) Finite Element Modeling of the Human Head at www.lib.kth.se/Sammanfattningar/kleiven020529.pdf. Larsen L.B, Larsen C.F. et al. (1991) Epidemiology of bicyclist’s injuries, IRCOBI Conference 217–230. McIntosh A. & Dowdell B. (1992) A field and laboratory study of the performance of pedal cycle helmets in real accidents, IRCOBI Conference, 51–60, IRCOBI Secretariat, Lyon, France. McIntosh A.S., Kalliaris D. et al. (1996) An evaluation of pedal cycle performance requirements, Stapp Car Crash Conference, SAE, 111–119. Mills N.J. (1990) Protective capability of bicycle helmets, Br. J. Sport. Med. 24, 55–60. Mills N.J. & Gilchrist A. (1991) The effectiveness of foams in bicycle and motorcycle helmets, Accid. Anal. Prev. 23, 153–163. Mills N.J. & Gilchrist A. (2003) Reassessing bicycle helmet impact protection, IRCOBI Conference 15–26.

350 Polymer Foams Handbook Mills N.J. & Gilchrist A. (2006a) Dynamic FEA of bicycle helmet oblique impacts, Int. J. Impact Eng. (submitted). Mills N.J. & Gilchrist A. (2006b) Bicycle helmet oblique impact tests, Int. J. Impact Eng. (submitted). Mills N.J. & Gilchrist A. (2006c) Bicycle helmet design, Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 220, 167–180. Peoplesize Software, Open Ergonomics Ltd, Loughborough, UK. Rivara F.P. et al. (1996) Report for the Snell foundation (on www.smf.org), Circumstances and severity of bicycle injuries. Smith T.A., Tees D., Thom DR. et al. (1994) Evaluation and replication of impact damage to bicycle helmets, Accid. Anal. Prev. 26, 795–802. Snell Memorial Foundation (1995) Standard for Protective Headgear for Use in Bicycling, New York 11780. Spaite D.W. et al. (1991) A prospective analysis of injury severity among helmeted and non-helmeted bicyclists involved in collisions with motor vehicles, J. Trauma 31, 1510–1516. Thompson D.C., Rivara F.P. & Thompson R. (2003) Helmets for preventing head and facial injuries in bicyclists, Cochrane Database of Systematic Reviews, Issue 4.

Chapter 15

Indentation, cracking, and fracture

Chapter contents 15.1 Introduction 15.2 Indentation 15.3 Plane strain indentation with a cube 15.4 Indentation with hemispheres 15.5 Crack growth in homogenous foams 15.6 Crack growth in bead foams Summary References

352 352 353 355 361 366 369 370


15.1

Introduction It is relatively easy to indent rigid polymer foams with a thumbnail. Although such damage is unsightly, fractures are of more concern – if a foam product breaks into pieces in an impact, its protective function will probably be lost. The cycle helmet case study showed that some expanded polystyrene (EPS) mouldings fracture in a brittle manner; bead boundaries are potential fracture paths, and their strength depends on the moulding conditions, so quality control is essential. The more expensive expanded polypropylene (EPP) foam is significantly tougher at room temperature. Alternatively EPS can be placed in a container with a high tensile strength (a cardboard box or a helmet shell) or moulded around a reinforcing structure with a high tensile strength. Cracks usually result from surface tensile stresses, which arise from bending or twisting loads, indentation, or oblique impacts on the surface. For instance, when a hemispherical striker impacts a flat block of EPS, the surface layer stretches near the periphery of the indentation and a pattern of circumferential cracks can develop. These cracks either arrest after penetrating a few bead diameters or they can cause catastrophic fractures. This chapter will deal with indentation before fracture.

15.2

Indentation Indenter geometries can be classified according to the strain field symmetry (assuming a flat-topped block of foam): 1. Cuboid: plane strain 2. Cylindrical: plane strain 3. Hemispherical: axisymmetric. Type 2 was introduced in Chapter 6 on finite element analysis (FEA), so emphasis will be on Types 1 and 3. During indentation by a hemisphere, the high strain region in the foam is hidden from view. In contrast, for plane strain indentation, the visible surface deformation field is the same as the internal field. Hence plane strain indentation may be preferred experimentally. However, if through-thickness cracking occurs, the foam separates into several pieces, whereas for axisymmetric indentation, the foam surrounding the indentation resists crack propagation. High-speed photography is necessary to reveal the maximum deflection during an impact; the permanent deformation after impact is not a good guide because closed-cell foams recover significantly from their maximum strain state.

Chapter 15 Indentation, cracking, and fracture

15.3

353

Plane strain indentation with a cube Stupak and Donavan (1991) compressed a sheet of EPS with a rectangular, flat-faced indenter (Fig. 15.1), and noted that cracks propagated in a plane at 35° to the applied stress axis. The main cracks (Type I) propagated from the indenter corners. Later in the indentation process, Type II cracks appeared in EPS of low toughness. Gilchrist and Mills (2001) impacted XPS blocks 50 mm high, 50 mm deep, and 160 mm wide, centrally with the flat face of a 50 mm metal cube (Fig. 15.2). Where the cube edges crossed the foam block, two cracks grew rapidly at an angle of about 45° from the vertical. The vertical projected length of the crack was almost the same as the indentation depth. As the crack did not accelerate, its growth must have been determined by the indention depth. The compressed region, in the lower part of the foam block, is approximately twice as wide as the indenter at the maximum deflection. The extent of this load spreading is determined by the crack angle to the vertical. The strain is small in the material to the sides of the crack. Gilchrist and Mills (2001) performed static FEA of the foam indentation using a relatively coarse mesh with elements of length 5 mm. In order to obtain predictions at large indentations, the base of the metal cube had to be tied to the foam, except at the element closest to the corner. With dynamic FEA (unpublished), it was possible to use frictional contact on the cube surfaces and a smaller mesh (1 or 0.5 mm). No special condition was necessary on the element next to the corner since a small over-closure (overlap of the foam and metal) occurs. For an elastic continuum, a stress singularity occurs at such an indenter corner; the stresses tend to infinity as the corner is approached. However, foams are inhomogeneous on the cell size scale (typically 0.5 mm), so stress singularities do not occur. Nevertheless, the predicted plastic strain is a maximum near the indenter

Type I θ = 35°

Type I Type II

Figure 15.1

Pattern of indentation and cracking in EPS under a rectangular indenter (Stupak and Donovan, 1991 © SAGE).


(a)

(b)

(c)

(d)

Figure 15.2

Video frames of indenting a block of XPS35 foam with a 50 mm cube at the time indicated (Gilchrist and Mills, 2001).

Mirror symmetry plane

Cube 1.5

1.0

Figure 15.3

0.5

Rigid flat table

Dynamic FEA prediction of 50 mm thick PS35 foam block shape after 35 mm indentation with a 50 mm cube with contours of the plastic strain magnitude PEEQ (Mills, unpublished).


355

3

Striker force (kN)

2.5

2 FEA 1.5

1

Expt

0.5

0

Figure 15.4

0

5

10

15 20 25 Foam deflection (mm)

30

35

40

Cube indenter force vs. deflection: data for PS35 foam (Gilchrist and Mills, 2001) compared with dynamic FEA prediction (Mills, unpublished).

corner (Fig. 15.3), and there is a high tensile stress in the foam just above this touching the sides of the indenter. Assuming that cracks initiate when the principal tensile stress exceeds a critical value, they should form at the indenter corner. In general, cracks grow in a direction perpendicular to the largest principal tensile stress. However, in ABAQUS 6.5 it is not possible to simulate crack growth. The slope of the predicted indentation force vs. deflection relationship (Fig. 15.4) decreases after 2 mm deflection when the foam yields. The predicted force is about 20% higher than the experimental force for deflections less than 20 mm due to the failure to consider crack growth. At large deflections, the experimental force is larger, as crack growth causes significant load spreading.

15.4

Indentation with hemispheres Impact tests with hemispherical strikers can evaluate the ability of foams to resist intrusion and absorb energy; this is relevant to headform impacts on automobile internal padding. Two cases will be considered. ‘Large’ hemispheres, with radius R greater than the foam thickness t, can never penetrate the foam by more than a radius, so the contact area always increases with penetration. ‘Small’ hemispheres, with R ⬍ t, create greater tensile strains at the sides of the indentation so there is a greater chance of foam cracking.

356 Polymer Foams Handbook 15.4.1 Large hemispheres impacting EPS foam 15.4.1.1

Experimental data

Mills and Moosa (1999) tested 50 mm thick EPS moulded tiles made from three types of beads. Table 15.1 gives the failure stresses and strains of the moulded surfaces measured in three-point bending. Data, for uniaxial impact compression of 50 mm cubes in a direction perpendicular to the tile moulded surface, were analysed to obtain the parameters σ0 and p0 of equation (11.4). A 5.33 kg hemispherical striker of radius 50 mm was dropped on to the centre of 130 by 130 by 50 mm samples supported on a rigid flat table. The loading part of the force vs. deflection traces, for a range of drop heights, almost follows a master curve (Fig. 15.5), showing that viscoelasticity hardly affects the response.

Table 15.1 Failure properties of EPS mouldings Polymer Code* AS AL B

Nominal density (kg m⫺3)

Bead size prior to moulding (mm)

80 80 80

Tensile failure strain (%)

0.5–0.9 0.9–1.5

Tensile failure stress (MPa)

2.5 1.6

Fracture toughness (kNm⫺1.5) 74 ⫾ 13 80 ⫾ 8 121 ⫾ 8

0.76 0.74

* Code: Manufacturers A and B, bead sizes small and large. 10 FEA

Striker force (kN)

8

6

4

2 0.15 0.25 0

Figure 15.5

0

5

0.5

0.75 1

1.25 1.5

10 15 20 Foam deflection (mm)

1.75 2

25

2.25

30

Force–deflection graphs for experimental drops (heights in m) for AL foam of density 80 kg m⫺3 (Mills and Moosa, 1999) compared with dynamic FEA for 2.25 m drop (Mills, unpublished).


357

Indentation and cracking were observed, in 2 mm thick vertical slices through the centre of the indentation, using transmitted light. The densification pattern (Fig. 15.6a) differs from that assumed in Figure 13.8, due to the density gradient (shown in Fig. 4.5), and hence compressive strength gradient through the moulding. The yielded zone (1) is of the same width as the surface indentation for a 0.5 m drop, but it becomes narrower than it is for larger drops, as the main crack (2) develops. The zone depth increases with the drop height, but the maximum impact energy is insufficient to make it extend to the lower specimen surface. For 50 kg m⫺3 density foams, it only touches the lower surface at the midpoint for the maximum impact energy. A single annular crack (2), linking surface bead boundaries, initiates if the impact energy exceeds 13 J. Once it has penetrated a few mm into the moulding, it propagates mainly across beads, but it does not propagate into the unyielded foam. Its length is 7 ⫾ 1 mm smaller than the yielded zone depth. Subsidiary annular cracks (4) initiate outside the main crack, later in the indentation process. Seen parallel to the striker motion direction (Fig. 15.6b), their spacing is between 1 and 4 beads. These cracks relieve the surface radial tensile stresses from the indentation. 15.4.1.2

Analysis

Figure 14.7 shows the assumed contact geometry between a flat rigid surface and the outer surface of a helmet of radius RH, supported on a spherical headform (Mills, 1990). This geometry is appropriate for a flat block of foam pressed between a large radius hemisphere and a rigid flat table. The zero load-spreading model has vertical lines of force in the foam contact zone, so the cross-section of

4 2

1 50 mm (a)

Figure 15.6

50 mm (b)

AL foam of density 80 kg m⫺3 after hemisphere impact from 2.5 m: (a) 2 mm thick vertical section: 1, yielded zone; 2, main crack; 3, lower surface of specimen; and 4, subsidiary crack and (b) indented surface: 1, cut plane seen obliquely; 2, main crack; and 4, subsidiary crack (Mills and Moosa, 1999).


B

Loading slope (kNm−1)

0.5

0.4

A

0.3

0.2

0.1

0

Figure 15.7

0

0.2

0.4 0.6 0.8 Initial yield stress (MPa)

1

1.2

Comparison of loading slope for hemisphere impact with prediction of equation (15.1) for EPS foam (Table 15.1).

the crushed foam is the same at all levels. Assuming that a constant foam yield stress σy and that the foam crushing deflection x is much less than the hemisphere radius R, the force F transmitted by the foam is F ⫽ 2πRσy x

(15.1)

This linear loading relationship has a slope proportional both to the foam yield stress and to the hemisphere radius. Orriger et al. (1984) analysed the impact of hemispherical and ellipsoid strikers on viscoelastic foams using the zero load-spreading model, but did not check the model experimentally. The slope of the approximately linear force deflection graphs is compared, as a function of the initial yield stress, with the prediction of equation (15.1) in Figure 15.7. For yield stresses ⬍ 0.6 MPa, the simple analysis is accurate within 10%. The response is dominated by the foam compressive yield; variations in the surface tensile failure strain or fracture toughness play a minor part. However, for the 80 kg m⫺3 density foams, where the effective gas pressure p0 is greater than 70% of the initial yield stress, equation (15.1) is increasingly in error.


359

Hemisphere Hemisphere 1.4 1.2 0.8

0.8

0.6

0.0 0.4

(a)

Figure 15.8

0.2

0.6 0.4

Rigid table

0.2

Rigid table

(b)

Dynamic FEA of a 50 mm radius hemisphere impacting a 50 mm thick EPS foam block: (a) shape at 10.4 mm indentation with contours of vertical compressive stress (MPa), (b) at 25 mm deflection with contours of the plastic strain magnitude PEEQ (Mills, unpublished).

Dynamic FEA (Mills, unpublished) of the indentation of 50 mm thick blocks of EPS foam AL of density 75 kg m⫺3 used the measured parameters σ0 ⫽ 0.89 MPa and p0 ⫽ 0.64 MPa. The predicted force vs. deflection relation (Fig. 15.5) initially matches the experimental data, but overestimates them at high deflections. The predicted slope is constant, suggesting that the non-linearity of the experimental responses results from the surface cracking and/or the density variation through the tile. FEA is more accurate than the use of equation (15.1), but it does not provide a simple relationship between the indentation force, the hemisphere radius, and the parameters of the foam material model. Only a few of the contours of the vertical compressive stress (Fig. 15.8a) are vertical, unlike the assumption of the Mills (1990) model. Load spreading occurs at the base of the foam to a radial distance greater than the foam/hemisphere contact radius. The maximum compressive stress occurs close to the indenter on the vertical symmetry axis of the indenter. From the contours of plastic strain magnitude (Fig. 15.8b), the greatest densification occurs just under the hemisphere centre, and the plastic zone hardly penetrates the lower surface of the block. This confirms the experimental observations in Figure 15.6a. Zhang et al. (1997) used LS-DYNA to predict the impact of a 22.2 kg hemisphere of radius of 63.5 mm on a 101 mm thick block of EPP foam of density 49 kg m⫺3. The predicted force deflection response was nearly linear with slope 0.20 kN mm⫺1 close to the experimental data. However, the area beneath the loading curve is 3.7 times the 222 J kinetic energy calculated from the striker mass and impact velocity of 4.4 ms⫺1, so some detail is incorrect.


30 mm (a)

Figure 15.9

35 mm (b)

Residual deformation and cracks (arrows) in: (a) EPS and (b) EPP foams sectioned after impacts from a 35 mm diameter hemispherical striker (Gilchrist and Mills, 1996).

15.4.2 Small hemispheres impacting EPS and EPP Gilchrist and Mills (1996) impacted EPS and EPP cycle helmet foam liners with a 35 mm diameter hemisphere. Measurement of the pressure distributions on the inner surfaces of the liners showed that there was less load spreading with the EPS than with the EPP liners. The crack locations (Fig. 15.9) indicate that the surface tensile stresses are largest at the sides of the hemisphere. EPS foam, although it partly recovers from compressive deformation, suffers permanent deformation of about 50% of the peak compressive strain (Chapter 16). In the EPP with 50% higher fracture toughness than the EPS, there is less tearing at the edges of the impact area and more recovery after the impact. To compare load-spreading ability, 25 mm thick sheets of foam, supported on a flat rigid horizontal table, were impacted with the 35 mm diameter hemispherical striker. When the striker penetrates the foam by more than 17.5 mm, the nominal area of crushed foam under the striker is A ⫽ 962 mm2. The mean indentation stress σ– ⫽ F/A, where F is the striker force, was used to calculate the indentation stress ratio H, defined in equation (9.14). Figure 15.10 shows that H is close to unity for EPS foam, whereas it rises to 1.4 at a mean strain of 35% for EPP foam. The cracking at the edge of the indentation in the EPS removes any shear stresses acting at that location. The higher H for the EPP indicates significant load transfer to foam surrounding the contact area and to a large area of the lower surface of the foam block. Hence, for an impact with a small diameter rigid object, the peak pressure on the inner surface of helmet with an EPP liner is lower that on the outer surface. The greater load-spreading ability of EPP, compared with EPS, is due to its higher ratio of fracture toughness to compressive yield stress.


361

Indentation stress ratio (H)

1.6 EPP

1.4 1.2 EPS

1 0.8 0.6 10

20

30 40 50 Compressive strain (%)

60

70

Figure 15.10 Comparison of the relative load spreading of PS and PP foams for a 35 mm hemispherical striker impact as a function of the strain in the impact (Gilchrist and Mills, 1996).

15.5

Crack growth in homogenous foams 15.5.1 Fast crack growth and fracture mechanics Linear elastic fracture mechanics (LEFM) describes the characteristic elastic stress field surrounding the crack tip in isotropic continuum materials (Mills, 2005). This is shown in Figure 15.11 for the common mode I type of loading, which causes crack opening. Modes II and III cause shear displacement of the crack faces. Cracks often change direction to propagate in mode I after initial growth in another mode. In the mode I stress field, the mean tensile stress σ– and maximum shear stress τmax (used to generate the isochromatic pattern in Fig. 15.11) are expressed in terms of polar coordinates (r, θ) as

σ ≡

σxx ⫹ σyy 2

KI

⫽

2πr

cos

θ 2

2

⎛ σyy ⫺ σxx ⎟⎞ ⎜ ⎟⎟ ⫹ σ 2 ⫽ KI sin θ τ max ≡ ⎜⎜ xy ⎟⎟ ⎜⎝ 2 2 2πr ⎠

(15.2)

Both stress components are zero along the crack surface as required. The scaling constant KI, known as the stress intensity factor, has the dimensions of stress times the square root of length. There can be a


4

2

y 5

z

6

Crack

1

r

θ x

6

1

5

4

3

2

Figure 15.11 Contours of the characteristic maximum shear stress field around a crack tip under mode I loading.

yielded zone very close to the crack tip surrounded by the LEFM stress field. The hypothesis of LEFM is: A crack will grow if and only if the stress intensity factor KI exceeds the fracture toughness KIC of the material. If the fracture can be described by LEFM, KIC should be independent of the crack length, the fracture surface should be flat on a macro-scale, and crack propagation should be at a high speed. Hence the loading force should fall suddenly as the crack propagates (except for some unusual specimen geometries). Conversely, if the force falls slowly as a crack propagates, it is unlikely that the fracture can be described by KIC . Two issues need to be addressed: 1. Does an LEFM stress field exist in a foam? Choi and Sankar (2005) modelled the fracture toughness of open-cell carbon foams, with a failure strain ⬍2%, using a simple-cubic array of edges. In Figure 15.12 of the deformation field around the crack, there is a band of shear running almost vertically from the crack tip because the foam model is highly anisotropic. However, if a nearly isotropic foam model were used, it should confirm that LEFM stress fields exist in foams made from materials with low failure strains. 2. Is the extent of the yielded region near the crack tip comparable with the specimen thickness? If the crack faces open by a significant angle and the crack advances by tearing through the yielded material, it is unlikely that the fracture can be treated by LEFM.


363

Figure 15.12 Model of a loaded crack tip in a simple cubic foam lattice with edge length 200 µm and edge width 20 µm (Choi and Sankar, 2005 © Elsevier).

It seems that LEFM describes the fracture of methacrylamide and rigid polyurethane (PU) foams, and EPS under impact loading. However, it cannot describe the fracture of semi-rigid low density polyethylene (LDPE) foam or flexible PU foams.

15.5.2 Models for the fracture toughness of open-cell foams Maiti et al. (1984) proposed a regular hexagonal honeycomb model, under tension in the plane of the cells. While this may be appropriate for ceramic or metal open-cell foams, it is unlikely to be relevant to flexible PU foams. The ‘crack’ passes through faces fractured where it crossed the negative x-axis (Fig. 15.13). When re-explained by Gibson and Ashby (1997), the crack ran through the model shown in Figure 7.5. They argued that a mode 1 elastic stress field exists from which the tensile loads on edge B can be computed. They assume, without justification, that the crack tip is located a half-edge length from edge B. (Intuitively it would appear to be at edge B, but then the stress in the edge would be indeterminate!) Choi and Sankar (2005) call this distance, from the crack tip to the nearest edge, the effective length. The assumed failure mechanism was the plastic bending collapse of the edges A and C, followed by the tensile failure of edge B. Large-scale deformation of the structure was not allowed. The plastic hinge mechanism, the honeycomb structure, and the assumption about the effective length led to the fracture toughness being proportional to the 1.5th power of the relative density. However, this model does not relate to open-cell PU foams. Choi and Sankar (2005) used FEA on the model shown in Figure 15.12, with uniform edges of square cross-section and a crack plane along a (100) plane. Assuming that the edges fail at a critical tensile stress, they predicted, for relative densities R in the range 0–0.05, that KIC is proportional to R (They give different relationships for R ranging up to 0.15 depending on whether the edge width b or the edge length L

364 Polymer Foams Handbook y

A C

B Crack

x

A C

Figure 15.13 Region around a crack tip in a loaded honeycomb (redrawn from Maiti et al., 1984).

is kept constant. This suggests some modelling problems as R is a unique function of b/L.). When they used Maiti et al.’s approach of computing the edge load from the KI field, they showed that the effective length was a fraction of the cell diameter, which increased with the foam relative density. This revealed a flawed assumption in Maiti et al.’s model. Neither of the above models is relevant to closed-cell foams made from polymers with a large failure strain. However, Maiti et al. (1984) claim the model in Figure 15.13 can be used. Using the appropriate relationship between the plastic bending moment and the face dimensions, they deduce that KIC should be proportional to R2.

15.5.3 Fracture toughness data for thermoplastic foams McIntyre and Anderson (1979) measured the KIC of rigid, predominantly closed-cell, PU foams of a wide range of densities. There was a linear relationship between KIC and for density ⬍ 200 kg m⫺3 (Fig. 15.14). At higher densities when the microstructure is unlikely to consist of polyhedral cells, the relationship becomes non-linear. They found that KIC was independent of the crack length in their single edge notch specimens. Maiti et al. (1984) tested brittle polymethacrylimide closed-cell foams of densities from 34 to 186 kg m⫺3, and found that the fracture toughness varied with the 1.42th power of density. No details were given of the fracture surfaces or failure mechanisms.


365

Fracture toughness KIC (MPa m0.5)

0.3

0.2

0.1

0

0

100

200 Density (kg m−3)

300

400

Figure 15.14 KIC vs. density for rigid PU foams (redrawn from McIntyre and Anderson, 1979 © Elsevier).

The fracture toughness of foams used as cores in sandwich panels (Chapter 18) has been studied because cracking weakens such structures. Zenkert and Bäcklund (1989) tested Divinylcell H200 PVC foam of nominal density 200 kg m⫺3 and found that the fracture toughness decreased slightly with increasing mean cell diameter (Fig. 15.15). Danielsson (1996) measured the fracture toughness of toughened PVC Divinylcell HD foams. Three-point bend specimens of two different span-to-depth ratios and centre-cracked sheet specimens were tested at a slow loading rate. The KIC value was found to increase almost linearly with foam density over the range 90–235 kg m⫺3 (Fig. 15.16). The specimen design affected the result to some extent (this could be due to different crack growth directions in the anisotropic foam). Viana and Carlsson (2002) found that the fracture toughness of Diab H foams increases linearly with density R in the range 60–300 kg m⫺3. Hence any model for the fracture toughness of closedcell foams should make a similar prediction. Kabir et al. (2006) showed that the fracture toughness of Divinylcell H130 foam was a function of the direction of crack growth for the same crack plane. When it was the flow direction the average KIC at a particular loading rate was 0.22 ⫾ 0.01 MN m⫺1.5, but when it was the rise direction KIC was 0.28 ⫾ 0.01 MN m⫺1.5.


Fracture toughness KIC (MN m−1.5)

0.6 0.5 0.4 0.3 0.2 0.1 0 0.2

0.25

0.3

0.35

0.4

Mean cell diameter (mm)

Figure 15.15 Fracture toughness of Divinylcell D foam vs. mean cell diameter (redrawn from Zenkert and Bäcklund, 1989 © Elsevier). 1.2 X

KIC (MPa m0.5)

1 0.8 X

0.6 0.4

X

X

0.2 0 50

75

100

125

150

175

200

225

250

Density (kg m−3)

Figure 15.16 Fracture toughness of Divinylcell HD foam vs. density, using centre-cracked tensile specimens (crosses) and cracked three-point bend specimens (circles) (Danielsson, 1996 © RAPRA).

15.6

Crack growth in bead foams 15.6.1 Fracture If crack growth follows weak bead boundaries in EPS mouldings, the resulting fracture surface consists of bead face facets, which do not

367

Chapter 15 Indentation, cracking, and fracture 120 29.2

25.9

23.3

Toughness J (J m−2)

100

80

60

40

20

0

0

0.1

0.2

0.3

0.4

Fraction of trans-bead fracture

Figure 15.17 J vs. fraction of trans-bead fracture surface for EPS of the densities (kg m⫺3) marked (redrawn from Stupak, et al., 1991 © SAGE).

lie in a single plane (Figs 15.19b and 15.20b). Due to the complex crack tip geometry, the local value of the stress intensity factor K may differ from the average value. The local density can also vary with position in the moulding, while both the cell face thickness near the bead boundaries and the bead-boundary tensile strength can vary with the process conditions. Stupak et al. (1991) investigated the toughness of EPS, of density between 23 and 30 kg m⫺3, as a function of the moulding temperatures and times, using slow three-point bend tests on centrally cracked bars. As the force–deflection response was non-linear, it was not possible to use LEFM; so they analysed the results with what they called the J integral. However they did not follow the proper procedure developed for elastic–plastic metals, which strain harden in the region around the crack tip. Instead they calculated the energy input to the point of maximum load (the instant of crack growth was not recorded), divided by the area of the fracture surface, which has dimensions J m⫺2. If fracture occurred at bead boundaries, the toughness was low (⬵10 J m⫺2). However if trans-bead fracture occurred through the foam cells, it was high (⬵100 J m⫺2). J varied almost linearly with the trans-bead fraction of the fracture surface (Fig. 15.17).

15.6.2 Impact fracture toughness Mills and Kang’s (1994) test rig (Fig. 15.18a) used 20 mm thick ‘Compact Tension’ specimens of size 50 by 50 mm. A falling mass with velocity 1.85 m s⫺1 was caught by a plate connected to the specimen;


60

Quartz load cell CT specimen

Electro magnet

Force (N)

40

20

Falling mass Foam

0 0

(a)

2 Extension (mm)

4

(b)

Figure 15.18 (a) Test rig for KIC and specimen shape, (b) force-distance traces for the impacts on PS compact tension sample (Mills and Kang, 1994).

after the inelastic collision it continues at 1.6 ms⫺1. Even the toughest foams only absorbed a small fraction of the total kinetic energy of 5.5 J. The impact force increased linearly with specimen deflection (Fig. 15.18b), so the fracture toughness KIC can be calculated using the formula KIC ⫽

FY

(15.3)

B W

where F is the maximum force, B and W are the thickness and width of the specimen, respectively, and Y is a polynomial function of the relative crack length C ⫽ a/W given by

(

)

Y ⫽ C 29.6 ⫺ 185.5C ⫹ 665.7C 2 ⫺ 1017C 3 ⫹ 638.9C 4 (15.4) Slow specimen loading rates caused slow tearing from the pre-crack, but impact loading in 5 ms caused rapid crack growth. The failure mechanism could change with strain rate from trans-bead, involving tensile yielding of the cell faces, to bead-boundary fracture, which is independent of strain rate. The failure forces were of the order of 50 N (Fig. 15.18b). After the peak, the force declined to 0 in 3 ms as the crack advances 20 mm; so the average crack velocity was 7 ms⫺1. This is much lower

369


Table 15.2

Fracture toughness of PS foams (Mills and Kang, 1994) Foam density (kg m⫺3)

Source

KIC dry (kN m⫺1.5)

KIC wet (kN m⫺1.5)

50 70 67 56 22

Helmet Helmet Helmet Box lid Box lid

85 ⫾ 10 109, 89 84, 90 99, 78 110, 87

81 ⫾ 3 112 65 19

than the 200⫹ ms⫺1 velocity of brittle crack propagation in solid polycarbonates (Gilchrist and Mills, 1987). For dry cycle helmet foams, KIC was in the range 80–110 kN m⫺1.5, independent of foam density for the limited range examined (Table 15.2). When specimens were immersed in water for 24 h, there was no significant change in the toughness. Further testing (unpublished) of a range of foam densities ρ from 20 to 85 kg m⫺3 showed a density effect, with the fracture toughness in kN m⫺1.5 given by KIC ⫽ 0.955 ρ1.119

(15.5)

The few measurements on low-density box lid foams suggest that 24 h of water immersion could weaken the material. An equilibrium water content in the box lid mouldings would only be reached after 250 h and longer for the helmet liner moulding.

15.6.3 Relating fracture toughness to fracture mechanisms Kang and Mills (1994) distinguished two types of inter-bead fractures – true bead-boundary fractures and those with fracture just below the boundary (Fig. 15.19c). For the first, the bead skin is seen on the fracture surface (Fig. 15.20b). For the second, fractured cells are seen (Fig. 15.20a); cells close to the bead boundary are weaker than those in the bead interior. Small islands of a foreign material on the bead skin were shown to contain calcium. Therefore weak skin boundaries may be related to weak calcium stearate layers used to prevent pre-foamed beads agglomerating, as well as to the processing conditions. Given that the bead-boundary tensile strength is a function of EPS processing conditions, it is unrealistic to expect that the fracture toughness should be a unique function of the foam density.

Summary Many ‘rigid’ polymer foams are prone to cracking when indented. The low tensile strength of bead boundaries at the surface of EPS mouldings


(a)

(b)

(c)

Figure 15.19 Sketches of the observed fracture mechanisms: (a) trans-bead fracture near the razor cut, (b) bead-boundary fracture, and (c) cell fracture close to the bead boundary (Mills and Kang, 1994).

1 mm (a)

(b)

Figure 15.20 SEM micrographs of EPS fracture surfaces: (a) density 20 kg m⫺3 with fractured

cells, (b) density 70 kg m⫺3 helmet with partially intact bead boundaries (Mills and Kang, 1994).

makes these particularly prone to indentation cracking. Consequently their surfaces may need to be protected by a load-spreading shell, for instance a cardboard box or bicycle helmet shell, which resists fracture. Nevertheless quality control is needed for bead-boundary strength to avoid EPS having a low fracture toughness. Fracture toughness is also important for foams used as sandwich panel cores.

References Choi S. & Sankar B.V. (2005) A micromechanical method to predict the fracture toughness of cellular materials, Int. J. Solid. Struct. 42, 1797. Danielsson M. (1996) Toughened rigid foam core material for use in sandwich construction, Cell. Polym. 15, 417–435. Gibson M.F. & Ashby L.J. (1997) Cellular Solids, 2nd edn., Cambridge University Press, Cambridge, UK.


371

Gilchrist A. & Mills N.J. (1987) Fast fracture of rubber toughened thermoplastics used for the shells of motorcycle helmets, J. Mater. Sci. 22, 2397–2406. Gilchrist A. & Mills N.J. (1996) Protection of the side of the head, Accid. Anal. Prev. 28, 525–535. Gilchrist A. & Mills N.J. (2001) Impact deformation of rigid polymeric foams: experiments and FEA modelling, Int. J. Impact Eng. 25, 767–786. Kabir Md.E., Saha M.C. & Jeelani S. (2006) Tensile and fracture behaviour of polymer foams, Mat. Sci. Eng. A 429, 225–235. Maiti S.K., Ashby M.F. & Gibson L.J. (1984) Fracture toughness of some brittle cellular solids, Scripta Met. 18, 213–217. McIntyre A. & Anderson G.E. (1979) Fracture properties of a rigid PU foam over a range of densities, Polymer 20, 247–253. Mills N.J. (2005) Plastics: Microstructure and Engineering Applications, 3rd edn., Butterworth Heinemann, London. Mills N.J. (1990) Protective capability of bicycle helmets, Brit. J. Sports Med., 24, 55–60. Mills N.J. & Kang P. (1994) The effect of water immersion on the fracture toughness of polystyrene foam used in soft shell cycle helmets, J. Cell. Plast. 30, 196–222. Mills N.J. & Moosa A. (1999) Impacts of hemispherical strikers on polystyrene bead foams, J. Cell. Plast. 35, 289–310. Stupak P.R., & Donavan J.A. (1991) The effect of bead fusion on the energy absorption of polystyrene foam, Part II. Energy absorption, J. Cell. Plast. 27, 506–513. Stupak P.R., Frye W.O. & Donovan J.A. (1991) The effect of bead fusion on the energy absorption of polystyrene foam, Part 1. Fracture toughness, J. Cell. Plast. 27, 484–505. Viana G.M. & Carlsson L.A. (2002) Mechanical properties and fracture characterization of cross-linked PVC foams, J. Sandw. Struct. Mater. 4, 99–113. Zenkert D. & Bäcklund J. (1989) PVC sandwich core materials: mode I fracture toughness, Comp. Sci. Tech. 34, 225–242. Zhang J., Lin Z., Wong A. et al. (1997) Constitutive modelling and material characterisation of polymeric foams, Trans. ASME J. Eng. Mater. Tech. 119, 284–291.

Chapter 16

Motorcycle helmet case study

Chapter contents 16.1 Introduction 16.2 Roles of components 16.3 Helmet test standards 16.4 Helmet design 16.5 Lumped-mass modelling of direct impacts 16.6 Helmet design by FEA 16.7 Helmet optimisation References

374 376 381 382 383 393 398 399


16.1

Introduction This chapter considers the functions of the components of motorcycle helmets, emphasising the role of foams. The impact responses of the main components are modelled before considering the whole helmet, and the validation of modelling by comparison with laboratory impact tests. Finally, helmet optimisation is considered. The design of motorcycle helmets has been explained by helmet testers (Newman, 1978, 1993) and manufacturers (Pollitt, 1982). They considered materials selection, and emphasised the role of Newtonian mechanics in the impact performance. Newman divided the design criteria into functional (shock absorbing capability, penetration resistance, abrasion resistance, retention, and reliability) and non-functional categories (low cost, good aesthetics, comfort, lightweight, and good thermal characteristics). Figure 16.1 shows a section through a full-face motorcycle helmet and the main components. Compared with bicycle helmets: (a) The thicker shell is more effective at spreading external loads to a large area of the skull. (b) The higher shell mass leads to force oscillations in impacts, with the foam acting as an elastic spring. (c) The helmet covers more of the head, so, for many wearers, its fit to the head shape is less precise. The comfort foam inside the helmet stabilises its position but reduces the ventilation of body heat. Three articles (Shuaeib et al., 2002) review a range of motorcycle helmet issues, while Part II republished some of the author’s analyses.

PS foam liner Shell

Comfort foam Edge beading Visor mount Lower liner Hanger plate Chin strap Buckle

Figure 16.1

Section of a full-face motorcycle helmet (Mills, 1996).

Chapter 16 Motorcycle helmet case study

375

Cartoons (Fig. 16.2; Newman, 1975) illustrate some constraints on the design process. The wearer’s field of vision must not be compromised, in spite of the need to protect the forehead. Retention systems can be improved, but the weight must be low and comfort is critical. Finally, cost is all important in certain sectors of the market. Helmet manufacturers usually design helmets to meet the performance requirements in national standards, rather than considering the statistics of motorcycle crashes, and trying to minimise the number of injuries. The head injury mechanisms are the same as for bicycle helmets (Chapter 14). There is inertia in changing national standards. Most countries require motorcycle helmets to be worn, so Transport Ministries play a role in determining performance standards.

Multilayered laminated concentric spheres

Small circular eye ports to minimise stress concentrations

Chin strap exceeds all known STDS

(a)

(b)

Anterior inferior posterior strap passes under crotch and attaches to rear of shell in latch

(c)

Figure 16.2

Cartoons showing the perspective of the shell designer, the retention system designer, and the manufacturer.

376 Polymer Foams Handbook The European Standard Regulation 22, controlled by Transport Ministries with private meetings of GRSP (group of experts on passive safety), is issued by the United Nations in Geneva. These arrangements are cumbersome and changes to the standard take years to implement. There is still a British Standard Committee, which interested parties can attend.

16.2

Roles of components 16.2.1 The shell A motorcycle helmet shell, typically 3–5 mm thick, is either an injectionmoulded thermoplastic or a pressure-moulded thermoset reinforced with glass (sometimes with added carbon or Kevlar) fibres. Its mass is typically in the range 0.7–1.7 kg. Since the shell is at a large radial distance from the head centre of gravity, the angular inertia of some full-face helmets is double that of the head (Mills and Ward, 1985). This increases the risk of roll-off if the retention system is not well designed and/or the helmet is a poor fit to the wearer’s head shape. The shell roles are as follows. 16.2.1.1

Absorbing impact energy

The shell bends (Mills and Gilchrist, 1994) when the helmet is impacted and the underlying foam deforms. The energy to deform the shell depends on the following: (a) Impact site: The energy is higher in regions of doubly convex curvature, like the crown. (b) The shell material and thickness: Thermoplastics absorb energy by viscoelastic deformation, while glass-fibre reinforced thermosetting plastics (GRP) have a lower elastic limit, and the cloth layers delaminate and/or the fibres fracture. (c) Shape of the impacting object: Convex rigid objects cause higher shell strains than flat objects. White delaminated areas appear after in some GRP shells after impacts on convex anvils, but are rare for impacts on flat surfaces. A rule of thumb is that 30–40% of the impact energy of a BS 6658 test is taken by the shell (Gale and Mills, 1985). 16.2.1.2

Distributing localised impact forces

If a helmet is not worn, an impact with a rigid convex object, like a kerbstone or a projecting corner of a truck, would cause localised high pressures on the skull, hence possibly a depressed skull fracture,


377

at a force less than that needed to cause brain injury by excessive acceleration. The shell distributes localised forces (Chandler et al., 1991) and prevents localised penetration of the foam liner; it causes a large volume of the liner to deform. The shell thickness is partly determined by the need to pass a penetration test, in BS 6658 (but not in Regulation 22); the tip of a conical steel indenter (60° point angle with 0.5 mm radius) with a 90 J impact energy must not touch the headform. However, there are very few impacts with metal spikes in real crashes. To pass impact tests onto a hemispherical anvil requires a stiff shell, hence a high shell thickness, unless the liner is unreasonably thick. 16.2.1.3

Preventing liner fracture

Most helmets use expanded polystyrene (EPS) liners, which have a low tensile strength, so are prone to fracture if bent (Chapter 15). A no-shell helmet would fracture in a high-speed oblique impact on the road, leaving the head unprotected. 16.2.1.4

Sliding on road surfaces

The stiff, convex-shaped shell slides along a road surface without an excessive frictional force (Glaister and Mortimer, 1982). Stones would dig into the foam of no-shell helmets and exert a high tangential force on the helmet (Mills and Gilchrist, 1994). High levels of rotational brain acceleration cause injuries, so it is necessary to limit the ‘frictional’ forces tangential to the helmet surface. A no-shell helmet could not pass the oblique impact test onto an abrasive surface in BS 6658. 16.2.1.5

Attaching components

Visor pivots and chin strap ends are attached to the shell. The dynamic force on a chin strap of helmets was as high as 3.5 kN when accident damage was reproduced in the laboratory (Glaister et al., 1979). Prior to 1980, failures initiated in polycarbonate (PC) shells where the chin strap was riveted to the shell. Chin straps are normally sewn to steel anchor-plates, which are riveted to the shell (Fig. 16.1). 16.2.1.6

Protecting the face and temples

Impacts on the face and jaw areas are common in motorcycle crashes (Otte, 1991). The area of the head protected by full-face helmets is significantly larger than that protected by an open-face helmet. However, the wearer’s field of view must not be restricted, and fullface helmets must not restrict head motion. Figure 16.3 shows an


Rotation

Crushed foam Rigid flat surface (road)

Figure 16.3

When the forehead of an open-face helmet hits a flat surface, the helmet can rotate allowing the head to impact the surface.The chin bar of a full-face helmet protects the face (Mills, 1996).

impact on the forehead of an open-face helmet where the contact area reaches the edge of the shell; the head and helmet rotate allowing the lower part of the forehead to strike the flat surface. Some open-face helmets do not have protective padding in the ear region, leaving this vulnerable part of the head with minimal protection. Full-face helmets often have shock-absorbing foam below the ear at the sides, and in the chin bar to meet the chin-bar impact test in BS 6658. This extra foam can prevent the helmet rotating in an impact with a flat surface, thereby protecting the cranium. The chin bar in full-face helmets has a double role: preventing facial bone fractures from direct blows on the chin and preventing the lower part of the forehead and temple being struck as the helmet rotates. The chin bar contains rigid foam to absorb energy. 16.2.1.7

Allowing a durable, coloured finish

Helmet aesthetics are vital for sales, and colour schemes can match those of the motorcycle or rider’s leathers. It is not possible to paint foams, whereas fibreglass shells can be painted. Self-coloured thermoplastic shells can be used with adhesively bonded coloured labels.

16.2.2 The foam liner Newman (1993) considered that an optimal design would: (a) Compress all the liners to the full extent – a compressive strain of about 80%. (b) Minimise rebound, by returning little of the input energy.


379

Table 16.1 Conditions for impacts in motorcycle helmet standards

Peak acceleration (g) Impact velocity flat anvil (m s⫺1) Impact velocity hemi anvil (m s⫺1) Headform mass (kg) Headform rotation

BS 6658

Snell

Regulation

Type A

M-95

22/05

FMVSS 218

300 7.5/5.3 7.0/5.0 5 No

300 150 J/110 J 150 J/110 J ⬎5 and ⬍6.5 No

275 7.5 7.5 (kerb) 3.1–6.0 Yes

400* 6.0 twice 5.2 twice 5.0 medium No

* Also requires ⬍2.0 ms of acceleration ⬎200g and ⬍4 ms of acceleration ⬎150g.

(c) Maintain constant head acceleration throughout the impact. This implies a rapid rise in acceleration at the start of the impact and a rapid decrease at the end. The first and third of these objectives are extremely difficult to achieve. In current designs the foam is only crushed near the impact site, and the amount of crushing depends on the impact velocity. Analysis later in this chapter shows that the impact force rises linearly with the foam compression. If the shell was infinitely rigid, it would increase the amount of foam that is compressed or sheared in a crash; however, such a shell would be very heavy. The main role of the liner is to provide a stopping distance for the head. A 30 mm thickness of foam can provide a maximum 27 mm stopping distance for the head, reducing its linear and rotational acceleration. Figure 14.10 and 14.12 showed how the foam compressive yield strength should be matched to the curvature of the impact site, the shape of the object impacted, and the impact energy to provide optimum protection. There is a physical limit to the liner thickness, because air resistance of the helmet must not cause neck muscle fatigue when riding at speed, which places an upper limit on the protection level. The ideal foam yield stress depends on the following: (a) Impact velocity component normal to the object struck: In BS 6658 impact tests occur at velocities in the range 5–7.5 m s⫺1 directly into fixed anvils. Although crash impact velocities can be higher, they are rarely perpendicular to immovable objects. Real crashes occur at a range of impact velocities, most frequently at relatively low velocities. (b) Type of object struck: The most common type of object is flat and rigid (Vallee et al., 1984), usually the road surface, with flat deformable (car panels), round deformable, and round rigid (roadside poles) objects being successively less common. BS 6658 requires impacts on a flat rigid anvil and on a 50 mm radius rigid hemisphere (Table 16.1). For deformable flat surfaces the foam should be of low yield stress to compensate for a large contact

380 Polymer Foams Handbook area, whereas for convex rigid objects the foam should be of high yield stress to compensate for the small contact area (Mills and Gilchrist, 1991). (c) Impact site: The most common sites struck are the front and sides of full-face helmets (Otte et al., 1984). The edges of the shell, around the vision area of full-face helmets or the lower edge of open-face helmets, are much less stiff than doubly convex areas such as the crown (Mills and Gilchrist, 1994), with flattish areas at the sides of full-face helmets having intermediate stiffness. To compensate for this, the mean foam density (so yield stress) is lowered at the crown – often by removing part of the foam or by using a softer foam insert. However, impacts on the helmet crown are rare in real crashes. (d) Impact velocity component tangential to the object struck: If this is high, the frictional forces on the shell rotate it relative to the road surface, bringing new areas of foam into the high-stress contact area. Analysis shows this has a minor effect on the required yield stress. Helmets cannot prevent all injuries, as some crashes are too severe for any wearable helmet. BS 6658 aims to protect riders from injuries in the majority of impacts. The impact tests are carried out in BS 6658 at ⫺20°C and at 50°C, and the compressive collapse strength of EPS decreases by about 20% over this temperature range. The majority of crashes occur with the helmet components in the 10–30°C range. The performance in the laboratory impact test would be somewhat better in this temperature range. The results vary from site to site and with the anvil, so the peak decelerations for a 7 m s⫺1 impact might be in the range 150–250g. The 30–35 mm thickness of liner foam provides a bending stiffness of the same order of magnitude as the shell. This aids load transfer from the contact area to remote parts of the skull, so reduces the peak compressive stress on the skull.

16.2.3 The comfort foams The comfort foams inside helmets are open-cell flexible polyurethane (PU) or polyvinylchloride (PVC) foams bonded to a cloth layer, typically 5–10 mm thick when uncompressed. Gilchrist et al. (1988) measured this gap while a large helmet was being worn. At local high points on the head the foam was compressed, while there were air gaps at other locations.

16.2.4 The retention system The human head varies in circumference and shape, while manufacturers from different countries use different shape liner mouldings.


381

Most helmet manufacturers sell four sizes of helmet, covering the ranges 53–54, 55–56, 58–59, and 60–61 cm head circumference. The length to breadth ratio of the helmet liner interior is usually the same for all sizes, often matching that of BS EN 960: 1995 headforms. There may only be two liner sizes, with the fit being improved by an internal headband, 25 mm wide band of soft foam, bonded to the interior of the liner at the level of the maximum head circumference, as well as the comfort foam. It is vital for the helmet liner shape to fit the head shape well, so the chin strap system can work effectively. The strap must be pulled as tight as is bearable against the soft tissue under the chin. BS 6658 advises tightly fastening the chin strap, then attempting to pull the helmet off forward by gripping the rear and pulling. This simulates helmet roll-off, which can occur when a rider hits the side or front of a car and somersaults over the bonnet. Measured decelerations on dummy heads during this stage are of the order of 20g, so the helmet is pulled upwards with a circa 200 N force, possibly causing roll-off, which leaves the head unprotected for any impact with the road (Mills and Ward, 1985). The retention systems effectiveness depends to some extent on the comfort and headband foams. These open-cell foams creep under the compressive stresses imposed by the chin strap, and other parts, where the sides of the head compress the foam, suffer stress relaxation. Therefore, helmet fit deteriorates a period of years of use. The cloth cover of the foam at the rear base of the helmet interacts with the nape of the neck, resisting helmet’s forward rotation.

16.3

Helmet test standards Motorcycle helmets standards (Table 16.1) require direct impacts. Some crash types are emphasised; since rigid flat objects are most frequently hit, flat steel anvils are used. In 1998, with Amendment 04 of UN Regulation 22, impacts onto hemispherical anvils were replaced by impacts on a kerbstone shaped anvil, and the need for a second impact on the same site was removed. Consequently, lower-density foam liners are now used. Impact sites are defined with respect to markings on headforms (Fig. 16.4). An oblique impact test was introduced in BS 6658 to test projections, such as metal bolts used as visor pivots, but it does not measure headform rotational acceleration. Later research showed that it limits the rotational acceleration to the correct order of magnitude. Metal or filled-plastic headforms are used, that resist damage and do not absorb energy, so the test results are reproducible. It is assumed that, if a human head and a headform undergo the same impact while wearing a helmet, the acceleration of the headform correlates with the strains in the human brain, and hence with the injury severity.

382 Polymer Foams Handbook P

Central vertical axis

B

12.7 mm

R A

20°

20° x

B

A⬘

x

Rear view

Figure 16.4

Regulation 22 impacts sites B front, R rear, X side, and P near crown on a helmet, relative to markings on the headform (redrawn).

However, the acceleration limit in standards (Table 16.1) may not minimise the number of head injuries; 300g may be too high – the limit used in car crash testing is 85g. Regulation 22 also requires that the head injury criterion (HIC) ⬍2400. This time-weighted integral of the acceleration trace is affected by the trace shape, so helmets with peak accelerations ⬍275g can have HIC ⬎2400. Thom et al. (1997) compared the performance of 36 helmet types which pass FMVSS 218. They found, for the same headform and impact sites, the mean peak linear acceleration using the freely rotating headform of Regulation 22 was 200g, whereas with a non-rotating headform (constrained by a rigid neck) it was 252g. The centre of gravity of the Regulation 22 headform was not in line with the impact site, so some of the impact kinetic energy was converted into rotational kinetic energy. Consequently, a constraint on headform rotation affects the test results.

16.4

Helmet design Modelling can assist in the design of helmets, and assist in the interpretation of acceleration–time traces from helmet impact tests. An understanding of the main deformation mechanisms of the helmet components, and the causes of mechanical vibration in an impact, allows the limitations of a helmet design to be evaluated. Lumpedmass models are considered first, because the component deformation


383

can be compared with direct measurements. In simulations, the impact type and constraints on the headform motion are usually one of the following: (a) Direct impact with no headform rotation, as in the Snell (2005) and BS Standards. These conditions produce the highest headform acceleration for a given impact velocity. Most finite element analysis (FEA) has used these conditions; there is no need to specify headform angular inertia, and helmet rotation on the headform is unlikely. (b) Direct impact with headform rotation allowed, as in Regulation 22. There will be angular acceleration of the headform. (c) Oblique impacts with headform rotation allowed. This is the most realistic, but no published FEA has considered it. Analysis of helmet shell damage by Hurt et al. (1976), Mills (1996), and Richter et al. (2001) showed that most impacts are oblique to the shell surface. In helmet test standards, and in most FEA, it is assumed that the tangential component of the impact velocity has no effect on the helmet peak linear deceleration. This was shown in Chapter 14 to be a reasonable assumption for bicycle helmets, but has not been confirmed for motorcycle helmets.

16.5

Lumped-mass modelling of direct impacts 16.5.1 Lumped-mass models for helmets The simple bicycle helmet impact model (Section 14.5.1) ignored the bending stiffness of the thin (0.5 mm) helmet shell, and the forces needed to accelerate the light (⬍0.5 kg) helmet. Equation (14.2) predicted a linear relationship between the impact force and the crushing deformation of the liner. However, significant forces are needed to accelerate motorcycle helmets of mass 0.8–1.5 kg. In lumped-mass models the motions of the centres of mass of the helmet shell and liner are considered; each mass has a single degree of freedom of linear motion along one axis. As helmet rotation is ignored, the moments of inertia are of no concern. Gilchrist and Mills (1987) used such a model to analyse crown impacts on industrial helmets, which have no foam liner. The parameters were as follows: (a) The elastic contact stiffness k between the helmet shell and the flat striker, in parallel with a viscous damper with constant n. (b) The effective mass me of the part of the helmet which moves at the velocity of the striker.


Helmet components and data for their impact behaviour Component

Data source

Data treatment

1 EPS crushing

Impact tests on rectangular blocks

2 EPS elastic deformation

Force difference between head and striker Impact site on shell with accelerometers attached Impact tests on rectangular blocks

Curve fit with equation (11.4), then integrate across contact area Measure resonant frequency and infer stiffness Data gives loading stiffness and force to buckle/delaminate Curve fit with equations (16.2) and (16.3)

3 Shell deformation

4 Comfort foam compression

The initial force peak was predicted to have magnitude F ⫽ V me k

(16.1)

where V is the impact velocity.

16.5.2 Helmet deformation mechanisms Table 16.2 lists the four main deformation mechanisms in a motorcycle helmet. Each mechanism appears as a spring/dashpot element in the model (Gilchrist and Mills, 1994). Figure 16.5a shows two load paths between the headform and the object struck. In the model (Fig. 16.5b) the links remain horizontal when the springs and dashpots deform. Load path 1 involves bending of the outer regions of the shell (parameters stiffness k1 and damping constant n1), in series with the elastic deformation of the liner (k2 and n2), and the comfort foam (k3 and n3). Load path 2 is direct through the crushed EPS, and the comfort foam. Position variables x1 to x4 are allocated to the centres of masses m1 to m4, which move along vertical x-axis. The parallel connection of two load paths implies that the anvil deforms the EPS from the outside. The area of crushed EPS is too small to allow the rigid headform to penetrate the inner surface of the liner. Consequently, the position variable x3 for the liner inner surface is the same for both load paths. Table 16.3 lists the component masses and their deformation measures. The component deformations are considered in the following sections.

385

Chapter 16 Motorcycle helmet case study Load paths 1

Load paths 1 2

2

Striker/anvil m1 k1a

Striker/anvil

n1 k1b Liner yield

l el

lin

er

k2

n2

am

rt

fo

f

m oa

Co m fo

Po lys ty

re ne

Sh

Shell m2

Liner m3 CL

k3

Headform

n3

Headform m4 (a)

Figure 16.5

Table 16.3

(b)

(a) Load paths between a rigid flat surface and the head and (b) the equivalent mass, spring, and damper model (Gilchrist and Mills, 1994). Parameters of lumped-mass model

Component

Mass

Striker or anvil Shell Liner

m1 m2 m3

Comfort foam Headform

Ignore m4

16.5.2.1

Deformation type

Deformation measure

Rigid Bend Compress Bend Compress Rigid

x3 ⫺ x1 x3 ⫺ x1 x3 ⫺ x2 x4 ⫺ x3

Characteristic time τ1 τ2 τ3

EPS crushing

Mills and Gilchrist (1994) used approximate methods to estimate the loading force–deflection relation for motorcycle helmets. The forces were summed on concentric foam annuli in the contact area; the contact force was found to rise linearly with deflection (Fig. 16.6). If the liner deflection at the centre of the contact area falls, the force is assumed to fall along a line of slope 5 MN m⫺1, an approximation to


Force (kN)

15

10

5

0

Figure 16.6

0

5


20

25

Predicted contact force between a flat surface and an EPS liner of radius 140 mm, thickness 25 mm, and density 56 kg m⫺3, as a function of the central deflection. An unloading path is arrowed (Gilchrist and Mills, 1994).

the slight positive curvature in experimental data (Gale and Mills, 1985). The reloading line follows the unloading line until the curve is met again. 16.5.2.2

Bending and shear of the EPS elastic region

During an impact on the crown of an open-face helmet, containing EPS of density 60 kg m⫺3, the difference between the headform force Fh and the striker force Fs oscillated at a frequency of 400 Hz (Gale and Mills, 1985). Using equation (9.1) for the resonant frequency ω of a mass–spring system, and the shell mass m2 ⫽ 0.6 kg, gives the liner spring constant k2 ⫽ 4.0 MN m⫺1. The Young’s modulus E for other EPS densities is given by equation (11.36). Equation (9.4) gave the compressive spring constant k, of a block of foam of area A, thickness t, and Young’s modulus E, as EA/t. There is assumed to be a similar relationship between spring constant and foam modulus for the helmet liner. Hence the liner stiffness k2 is proportional to the 1.6th power of the foam density. The dashpot constant was chosen empirically so the characteristic time τ2⫽ n2/k2 ⫽ 0.25 ms. 16.5.2.3

Shell deformation

At the impact site, 120 mm above the visor top, the shell radius of curvature was 120 mm laterally, and 140 mm in the fore and aft direction. Two miniature accelerometers, attached to the shell exterior


387

aT x

z

Striker

y

Shell EPS liner

Fixed headform

aL

Figure 16.7 Table 16.4

aR

Linear accelerometers on shell exterior used to measure shell bending (Gilchrist and Mills, 1994). Data from loading curves for deformation of full-face shells Shell material

Striker

k1a (kN m⫺1)

ABS ABS GRP GRP

Hemi Flat Hemi Flat

690 700 1350 2250

Event Buckle None Delaminate None

At force (kN) 3.0 3.5

remote from the impact site (Fig. 16.7), monitored the position of the shell centre of gravity. Averaging their accelerations cancels the effect of small helmet rotations. The mean acceleration was subtracted from the 5 kg striker acceleration, and then integrated twice with respect to time, to give the contact point deformation relative to the shell centre of gravity. The shell stiffness kla is the slope of a graph of striker force vs. the contact point deformation (Table 16.4); the thermoplastic helmet shells are less stiff than GRP ones. However, this ignores the EPS liner contribution to the effective shell stiffness. The thermoplastic shells buckled in impacts with the 50 mm radius hemispherical anvil, but not with the flat anvil. In Figure 16.8, the slope of the loading graph falls from kla 艑 700 to klb 艑 200 kN m⫺1 when the force exceeds the buckling force Fb. In the model, unloading is assumed to occur at slope kla until the force has fallen by Fb. On subsequent reloading the force increases at the slope kla until it rejoins the earlier loading line. Buckling appears not to damage thermoplastic

388 Polymer Foams Handbook 7 6 5 Force (kN)

Model 4 3

Experiment

2 1 0

Figure 16.8

0

5


20

25

Force vs. deflection trace for ABS shell of a full-face helmet hit by a hemispherical striker, compared with the model approximation for loading at 5 m s⫺1 and unloading at ⫺5 m s⫺1(Gilchrist and Mills, 1994).

shells. However, when GRP shells hit a hemispherical anvil, delamination between layers of the glass matt/roving/cloth reinforcement decreases the shell stiffness on reloading. The characteristic time for shell bending was chosen empirically as τ1 ⫽ 0.2 ms. 16.5.2.4

Comfort foam compression

The comfort foam response was measured at a low-strain rate, since it was difficult to measure the impact response of such soft foam. Figure 16.9 shows that the force rises rapidly above a strain of 80%. For modelling the contact area was taken as being constant and equivalent to a disc of radius 75 mm. The experimental curve was fitted by F34 ⫽ k3 (x4 ⫺ x3 ) ⫹ n3 (V4 ⫺ V3 )

if ε ⬍ 0.6

(16.2)

and, if the foam strain ε ⬎ 0.6, the factor k3 is multiplied by exp(40(ε ⫺ 0.6)2 )

(16.3)

to simulate the bottoming-out of the foam. The characteristic time for the comfort foam τ3 ⫽ 1 ms produces a viscoelastic effect when the loading velocity is changed (Fig. 16.9).


389

1.0

Force (kN)

0.8

0.6 5 0

0.4

−5 0.2

0.0 0

20

60

40

80

100

Strain (%)

Figure 16.9

Force vs. strain for 5 mm thick comfort foam, for a contact area of radius 75 mm, fitted with equations (16.2) and (16.3) using k3 ⫽ 40 kN m⫺1 and n3 ⫽ 40 Ns m⫺1. Curves labelled with the compression velocity m s⫺1 (Gilchrist and Mills, 1994).

16.5.3 Computer model The programme repeats the following sequence at time intervals of 2 µs: (i) The old positions xi and velocities Vi of the four masses are used. (ii) The forces are calculated; for instance that between masses 3 and 4 is F34 ⫽ k3 (x4 ⫺ x3 ) ⫹ n3 (V4 ⫺ V3 )

(16.4)

where x4 ⫺ x3 represents the comfort foam compression (Table 16.2). Details of the unloading responses are given in Sections 16.5.2.1 and 16.5.2.3. The forces on masses m1 and m4 cannot be negative, as the helmet can move freely away from the headform and the anvil. (iii) The accelerations are calculated using Newton’s second law. For instance a1 ⫽

F12 ⫹ F13 m1

(16.5)

(iv) The new velocities are calculated after a time interval ∆t from the old velocities using

(

)

Vi, new ⫽ Vi, old ⫹ 0.5 ai, new ⫹ ai, old ⌬t

(16.6)


100

80 Deflection (mm)

15

EPS elastic

60

10 40 EPS crush

5


Comfort foam

20

0

0

2

4

6 Time (ms)

8

0 10

Figure 16.10 Comfort foam strain and EPS liner crush and elastic deflections (zero shifted up by 10 mm) vs. time, for a flat striker impact at 7.0 m s⫺1 on an ABS shell with a 56 kg m⫺3 density liner (Gilchrist and Mills, 1994).

(v) The new positions are calculated by a similar numerical integration of the velocities over the time interval ∆t. Willinger et al. (2000) used the same lumped parameter model, and related it to a particular model of head vibrations. The helmet component deformations or strains are shown as a function of time in Figure 16.10, for an impact of a flat striker on a region of the shell with a 140 mm mean radius of curvature. The comfort foam compression rises rapidly to more than 90% of its thickness in the first 2 ms of the impact event, and remains close to 100% throughout most of the impact sequence. The striker force begins to rise during the initial comfort foam compression. The liner elastic bending is slightly negative while the comfort foam is being compressed; its inner surface below the impact site approaches the headform, while its outer regions have not moved. After 2 ms the bending becomes positive, as its outer parts move more than the contact region. Since the liner sides do not contact the headform, the shell mass m3 undergoes damped oscillations on the elastic part of the liner. The liner crush increases to a peak of 5 mm after 1 ms, then to a second peak of 12 mm after 4 ms. Figure 16.11a shows the predicted variation of the striker and headform forces with time. The striker force peak A occurs while the comfort foam is being compressed and the helmet is accelerating away from the striker. The first headform force peak B is due to an impact with the rapidly moving liner. The subsequent minimum in the headform force at C and maximum at D are due to oscillation of the shell mass on the liner spring. The peak

Chapter 16 Motorcycle helmet case study 20

391

20 Head E

B D

10 C

Striker A 5

10 Striker 5

0 0

Head

15 Force (kN)

Force (kN)

15

2

4 6 Time (ms)

(a)

8

0

10

0

2

(b)

4 6 Time (ms)

8

10

Figure 16.11 Headform and flat striker forces for crown impacts on an open-face helmet at

7 m s⫺1: (a) predictions for same conditions as Figure 11.9 and (b) experimental data (Gilchrist and Mills, 1994). 20

5 0

(a)

10

0

2

4

6

Velocity (m s−1)

8

0 10

20 15 10

10

5 0

(b)

20

0

2

4 6 8 Velocity (m s−1)

Maximum liner deflection (mm)

10

Headform force (kN)

15

25 Maximum liner deflection (mm)

Headform force (kN)

20

0 10

Figure 16.12 Predicted peak headform force and liner deflection vs. impact velocity of the

crown of an ABS shell, radius of curvature of 140 mm and 56 kg m⫺3 EPS liner, on: (a) flat and (b) hemispherical anvils (Gilchrist and Mills, 1994).

striker force at E occurs close to the time where the liner compression is a maximum. The experimental data (Fig. 16.11b), for an equal energy impact on the crown of an open-face helmet, shows same features as in Figure 16.11a but the peak forces are smaller, probably due to some crushing of the liner from the inside by the headform.

16.5.4 Predicted effects of impact velocity on helmet performance Figure 16.12a shows how the peak headform force, in a falling headform test onto a flat surface, increases nearly linearly with the impact velocity until it reaches the failure limit of 15 kN (equivalent to the 300g acceleration limit in BS 6658: 1985) at a velocity of 6 m s⫺1 when the liner crush is only 44% of its thickness. Figure 16.12b shows, for impacts into a hemispherical anvil, the 15 kN limit reached

392 Polymer Foams Handbook when the velocity is 8.8 m s⫺1 and the liner is 98% crushed. Hence the 56 kg m⫺3 foam density is optimal for the hemispherical anvil, but too high for the flat anvil. At the front of the helmet, where the average shell radius of curvature is 130 mm, the shell stiffness is lower, due to the proximity of a free edge. For this impact site, a 56 kg m⫺3 density EPS will have a lower than optimal yield stress for the hemispherical impact, and a near-optimal yield stress for the flat impact. At the side of the helmet the average radius of curvature is 170 mm, and the foam yield stress is too high for impacts into a flat surface. Hence, it is impossible to have an optimal foam liner of a single density and uniform thickness, even if the impact velocity is fixed. If EPS of 32 kg m⫺3 density is used in simulated impacts with a flat anvil, the peak headform accelerations can be reduced by 27% for impact velocities up to 8 m s⫺1. The critical velocity to cause a peak headform force of 15 kN is increased by 25%, compared with using a foam of density 56 kg m⫺3.

16.5.5 The effect of a second impact Pedder (1993) found that multiple impacts do not occur on the same helmet site in crashes – they involve different sites as helmets rotate between the impacts. In Regulation 22/03, there was an impact at 7 m s⫺1 onto a flat anvil, followed by an impact at a site 15 mm away from the first, at 6 m s⫺1, onto a hemispherical anvil. The second impact site is insufficiently far from the first to avoid the prior damage. Mills and Gilchrist (1994) assumed that the impacts were at the same site, and predicted (Fig. 16.13), for a helmet with a 56 kg m⫺3 liner 20

Force (kN)

15

10 First impact Second impact

5

0

0

5

10

15

20

25

30

Deflection (mm)

Figure 16.13 Anvil force vs. helmet deflection for ABS shell and 56 kg m⫺3 EPS liner: first impact at 7 m s⫺1 onto a flat anvil and second impact at 6 m s⫺1 onto a hemispherical anvil (Gilchrist and Mills, 1994).


393

density, that the maximum helmet deflection second impact is 4 mm greater than in the first. Hence it is necessary to keep some liner crush distance in reserve, by using a foam of higher than optimal yield stress for the first impact, to pass the critical second test. The second impact was removed from Regulation 22/04, but still occurs in American Standards (Table 16.1). Modelling predicted that the second impact in BS 6658 was not critical, so there is not the same need to change the British Standard.

16.6

Helmet design by FEA 16.6.1 FEA predictions for direct impacts Although helmet impacts are best considered by dynamic FEA, there have been few published articles. One problem is that fibre reinforced shells are elastically anisotropic, with complex failure mechanisms. Another may be difficulties with the model for the EPS. The geometry of the head relative to the helmet interior, the interfacial friction conditions, and the material properties are all critical. Unless the FEA is validated, suggestions for helmet design changes, based on FEA predictions, are unlikely to be implemented. For example, Yettram et al. (1994) made unrealistic predictions using DYNA-3D to model crown impacts of open-face helmets on a flat rigid surface. Although their EPS impact compression data was reasonable, the GRP shells were assumed to be linearly elastic with Young’s moduli 20, 40, 60, or 90 GPa (all too high). The coarse FEA mesh, with a single linear element through the foam thickness, could not accurately simulate the liner deformation. Since the EPS liner exactly fitted the headform, with a ‘high’ interface friction coefficient, a high liner shear strain was predicted at the helmet sides, at the peak of an impact at 6.7 m s⫺1, when it should be small or zero. Consequently the predicted peak headform acceleration of 507g, for a 57 kg m⫺3 density EPS liner in a PC shell, is too high – such a helmet should pass the BS 6658 impact test. They also ignored the need to pass impacts on a hemispherical anvil. Hence their conclusions, that helmet liners are too dense and that lower modulus materials such as high-density polyethylene (HDPE) should be used for shells, are both wrong. Brands et al. (1997) used MADYMO to model a full-face helmet hitting a flat rigid surface under Regulation 22 conditions. The comfort padding was modelled as a space between the liner and the headform, and a ‘low’ friction coefficient was assigned between the rigid headform and the comfort foam. The helmet shell inner surface was unrealistically tied to the liner, simulated with a coarse mesh. They ‘fine-tuned’ the parameters for EPS of density 59 kg m⫺3; the EPS yield stress of 0.32 MPa was half of the expected value and Young’s modulus of 1.8 MPa was 10% of the expected value. This arbitrary

394 Polymer Foams Handbook adjustment of material parameters predicted peak head accelerations, for frontal and rear impact sites, within 3% of experimental results, at the expense of delaying the response to longer times than those observed experimentally (Fig. 16.14), and preventing FEA simulation of a lateral impact because of excessive headform rotation in the helmet. Their interpretation of the peaks in the headform force vs. time trace in a crown impact, confirmed that of Mills and Gilchrist (1994). Chan et al. (2000) successfully modelled impacts into the chin bar of full-face helmets, as in the Snell Standard. However they used an unrealistic initial head position, with the chin contacting the chin bar. Kostopoulos et al. (2002) used FEA for direct crown impacts onto a hemispherical anvil, in which the headform did not rotate. The shell was a 2 mm thick fibre composite, over a 35 mm thick EPS liner of density 50 kg m⫺3, and the headform exactly fitted the liner interior. Thus, the impact site and anvil shape were atypical of real crashes. Changing the fibre type had only a small effect on the predicted acceleration–time trace. The high-strain region in the liner was local to the hemispherical anvil, with very little deformation in remote parts of the helmet (Fig. 16.15). There was no comparison with experimental data. The author performed dynamic FEA of impacts on an open-faced helmet. The EPS liner of density 55 kg m⫺3 was modelled as a crushable foam with σ0 ⫽ 600 kPa and an effective gas pressure p0 ⫽ 130 kPa. It was 30 mm thick at the top and 24 mm thick at the sides. The 4 mm

Resultant head acceleration (g)

250

200

Mills FEA Simulation Experiment

150

100

50

0

0

2

4

6

8

10

12

Time (ms)

Figure 16.14 Headform acceleration traces for a frontal impact at 7.5 m s⫺1 on a flat anvil: predicted vs. experimental for a full-face helmet (Brands et al., 1997) plus Mills (unpublished) FEA for an open-face helmet.


395

thick acrylnitrile–butadiene–styrene (ABS) shell was simulated by shell elements, while the ABS had a yield stress of 50 MPa, hardening at higher strains. PU comfort foam was simulated in the lower part of the helmet sides and rear, and a typical chin strap position was used (Fig. 16.16). The rigid headform had mass 4.26 kg, as in Chapter 14.

Figure 16.15 Predicted deformation at the peak of a crown impact of a full-face helmet on a hemispherical anvil (Kostopoulos et al., 2002 © Elsevier).

Shell

EPS liner Comfort foam

Chin strap

Figure 16.16 Simulation of an open-face motorcycle helmet (Mills, unpublished).


Force (kN)

Head

10

0.4 0.2 0.6

Road 5 1

0 (a)

0

1

2

3 4 5 Time (ms)

6

7

8

2

3

(b)

Figure 16.17 Direct impact of an open-face helmet crown on a flat anvil: (a) time variation of forces and (b) compressive stress contours (MPa) on the liner outside at 3.5 ms and contact area (white circle) (Mills, unpublished).

For a crown impact on a flat anvil, the predicted anvil and headform forces (Fig. 16.17a) are similar to the experimental data in Figure 16.11b, and to the predictions of the lumped parameter model. The vertical compressive stress is spread by the shell to areas of liner well outside the contact area (Fig. 16.17b). It is no longer helpful to plot the anvil or striker force vs. the helmet deflection for this site, since force oscillations hide the underlying trend line. For other impact sites, oscillations in the headform force are much smaller. Hence impacts on the crown cause atypical responses. The performance at frequently impacted sites is more relevant. For a frontal site, the predictions in Figure 16.14 are close to Brands’ experimental data (although the headform mass is 24% smaller than his). For impacts at the side of the helmet, the shell on that side compresses the comfort foam and thereby limits the helmet rotation on the head. The peak headform linear acceleration varies with the impact site and anvil (Table 16.5), which is typical of helmet testing. It is necessary to ‘over design’ for some impacts, in order to pass other more critical tests in the standard. A crown impact on a flat anvil requires a foam liner with a lower yield stress, in spite of it being a rare impact site. Impacts on hemispherical anvils produce higher liner deflections, hence require a higher foam yield stress. This questions the benefit of using a hemispherical rather than a kerb anvil, given the influence of the former on helmet design; the most common types of crashes should have the greatest influence. These simulations reveal limitations in the modelling of the unloading and reloading of crushed regions of EPS. The ‘excessive’ oscillations in the headform force, predicted for crown impacts, appear related to the lack of energy absorption mechanisms in this EPS.


397

Table 16.5 Simulated impacts of an open-face helmet Site

Anvil

Velocity (m s⫺1)

Peak head acceleration (g )

Front 70°

Flat Kerb Hemi Flat Kerb Hemi Flat

7.5 7.5 7.0 7.5 7.5 7.0 7.5

215 171 146 224 178 144 358

Right 80°

Crown

Peak head rotational acceleration (k rad s⫺2) 3.5 2.5 2.0 8.9 8.3 7.6 9.4

Peak liner compression (mm) 20.7 23.5 24.8 14.9 16.9 20.4 12.3

Predictions depend on the headform fit to the liner interior, since contact between these affects load spreading to the liner at the sides of the impact site. Motorcycle helmets are usually tested with headforms of the same shape as the liner interior, but slightly smaller. As human heads vary considerably in shape and size, they are less likely to be a good fit to the liner interior at the impact site. The predictions also depend on the retention system; if this is effective at preventing helmet rotation, the peak headform linear acceleration is slightly higher.

16.6.2 Validation It is rare that helmets are tested over a range of impact velocities, varying the anvil and impact sites. Zellmer (1993) impacted full-face motorcycle helmets, with PC shells and EPS liners of unspecified density, under Regulation 22 conditions, with a 60 cm circumference headform of mass 5.6 kg. His results for a frontal impact site are compared with FEA predictions for a 4.26 kg headform in an openface helmet in Figure 16.18. The slope of the trend lines are the same, but the predicted head accelerations are higher by about 25g. As the foam density in the tested helmet is unknown, the agreement may be fortuitous. The FEA predictions suggest that the 30 mm liner is about to bottom out at a 9 m s⫺1 impact velocity. Beusenberg and Happee (1993) analysed headform acceleration vs. time traces from Regulation 22 type impacts, when the helmet forehead hit a fixed flat anvil. They showed that, for a variety of PC and GRP shelled helmets, the headform force rose almost linearly with the displacement of the headform relative to the helmet, and the loading slope was higher for GRP helmet shells. For the latter, the headform crushed the foam liner from the inside, rather than the pattern of deformation shown in Figure 14.7. Mills and Gilchrist (1992) made a similar observation. The surface of metal headforms does not

250

25

200

20

150

15

100

10

50

5

0

0

2

4

6

8

Liner crush (mm)

Peak resultant head acceleration (g)


0

Impact velocity (m s−1)

Figure 16.18 Peak headform acceleration for frontal impacts (site B, Fig. 16.4) on a flat surface: circles, Zellmer’s experiments for full-face helmets; triangles, Mills (unpublished) FEA for open-face helmets; and crosses, liner crush (right-hand scale).

exactly fit the inside of the liner, so local pressure high spots occur in an impact. However, it is rare for internal liner crushing to be observed in helmets recovered from crashes, whereas external liner indentations are common. The skull, scalp, and hair and comfort foam all deform, spreading the load on the inside of the liner and preventing local indentation of the liner interior. It also needs to be proved that the mass of the metal headform should equal the total head and neck mass. In an impact, the soft tissue external to the skull and the articulating neck will deform, so the effective mass of the head and neck will be considerably smaller than its total mass. One way to determine the effective mass would be to perform impacts with cadaver heads in motorcycle helmets, and to compare the peak head accelerations and helmet liner deformations with tests using metal headforms of a range of masses. FEA is preferred to the lumped parameter model, as there is no need to experimentally determine a number of helmet mechanical parameters. Nevertheless, the success of the model in predicting events in a crown impact shows that these have a relatively simple explanation.

16.7

Helmet optimisation If motorcycle helmets were optimised for impacts on to rigid flat surfaces, the liner foam density would be lower than that used at present. Gilchrist and Mills’ modelling (1994) suggested changes in test standards, many of which occurred in Amendment 04 of Regulation 22,


399

brought into force in 1998. They suggested that only a single impact test should be performed at a site, and the hemispherical anvil impact should be abandoned or replaced by an impact with a kerbstone. The contact area with a kerbstone anvil in an impact is larger than with a hemispherical anvil, so lower-density foam liners can be used. It was also suggested that the penetration test, using a conical indenter, lead to shells being too thick. Surveys by Pedder (1982, 1993) show that shell penetration by sharp objects is extremely rare. If a stiff site, like the crown or rear of a helmet with a GRP shell impacts a flat anvil, the PS foam density must be reduced to pass the test. Arai (www.araihelmet-europe.com) says it uses PS foam liners of densities about 34 kg m⫺3 with high stiffness GRP shells. They use a slightly higher density in the forehead area, to compensate for the lower radius of curvature of the shell in this region. It is impossible to optimise the design of helmets for all impact velocities and sites. The compromise design should attempt to minimise the total harm to helmet wearers. Liu et al. (2005) reviewed epidemiological surveys and concluded that helmets reduce the risk of head injury by around 72%, but that there was insufficient evidence to compare the effectiveness of different types of helmet.

References Beusenberg M.S. & Happee R. (1993) An experimental evaluation of crash helmet design and effectiveness in standard impact tests, IRCOBI (International Research Council on Biokinetics of Impacts) Conference, Eindhoven, pp. 307–323. Brands D.W.A., Thunnissen J.G.M. & Wismans J.S.H.M. (1997) Modelling head injury countermeasures: a 3D helmet model, AGARD CP-597, Paper 26. BS 6658 (1985) Protective Helmets for Vehicle Users, British Standards Institution, London. BS EN 960 (1995) Headforms for Use in the Testing of Protective Helmets, British Standard Institution, London. Chan C.H. et al. (2000) FEA of helmet chin bar performance, J. Biomech. Eng. 122, 640–646. Chandler S., Gilchrist A. & Mills N.J. (1991) Motorcycle helmet load spreading performance for impacts into rigid and deformable objects, IRCOBI Conference, Bron (France), pp. 249–261. Gale A. & Mills N.J. (1985) Effect of polystyrene foam density on motorcycle helmet shock absorption, Plast. Rubber Proc. Appl. 5, 101–108. Gilchrist A. & Mills N.J. (1987) Construction-site workers helmets, J. Occup. Accid. 8, 199–211.

400 Polymer Foams Handbook Gilchrist A. & Mills N.J. (1994) Modelling of the impact response of motorcycle helmets, Int. J. Impact Eng. 15, 201–218. Gilchrist A., Mills N.J. & Khan T. (1988) Survey of head, helmet and headform sizes related to motorcycle helmet design, Ergonomics 31, 1395–1412. Glaister D.H. & Mortimer P. (1982) A test for the sliding resistance of protective helmets, RAF Institute of Aviation Medicine, Farnborough, Hants, Div. Record 28. Glaister D.H., Duff A. et al. (1979) Dynamic testing of helmet chin straps, RAF Institute of Aviation Medicine, Farnborough, Hants, Report 579. Hurt H.H., Oullet J.V. & Wagar I.J. (1976) Analysis of accident involved motorcycle safety helmets, Proceedings of the 20th Conference of the American Association of Automotive Medicine, Atlanta, pp. 187–197. Kostopoulos V., Markopoulos Y.P. et al. (2002) FEA of impact damage response of composite motorcycle safety helmets, Composites Part B 33, 97–107. Liu B., Ivers R. et al. (2005) Helmets for preventing injuries in motorcyclists, The Cochrane Library, Issue 1. Mills N.J. (1996), Accident investigation of motorcycle helmets, Impact 5, 46–51. Mills N.J. & Gilchrist A. (1991) The effectiveness of foams in bicycle and motorcycle helmets, Accid. Anal. Prev. 23, 153–163. Mills N.J. & Gilchrist A. (1992) Motorcycle helmet shell optimisation, Proceedings of the Association for the Advancement of Automotive Medicine, Portland, OR, pp. 149–162, AAAM, Des Plaines, IL. Mills N.J. & Gilchrist A. (1994) Impact deformation of ABS and GRP helmet shells, Plast. Rubber Comp. Proc. Appl. 21, 151–160. Mills N.J. & Ward R. (1985) The biomechanics of motorcycle helmet retention, IRCOBI Conference, Gothenburg, 117–127. Newman J.A. (1975) On the use of the HIC in protective headwear evaluation, Proceedings of the 19th Stapp Car Crash Conference, San Diego, pp. 615–639. Newman J.A. (1978) Engineering considerations in the design of protective headgear, Proceedings of the 22nd Conference of the American Association of Automotive Medicine, Ann Arbor, pp. 278–293. Newman J.A. (1993) Biomechanics of human trauma, head protection, in Accidental Injury, Eds. Nahum A.M. & Melvin J.W., Springer Verlag, New York, pp. 292–310. Otte D. (1991) Technical demands on safety in the design of crash helmets, 35th Stapp Car Crash Conf., 335–348, S. A. E., Warrendale, PA. Otte D., Jessl P. & Suren E.G. (1984) Impact points and resultant injuries to the head of motorcyclists, IRCOBI Conference, Delft, pp. 47–64.


401

Pedder J.B. (1993) Characteristics of serious and fatal motorcycle accidents. Ph.D. thesis, University of Birmingham. Pedder J.B., Hagues S.B. & Mackay J.M. (1982) Head protection for road users with particular reference to helmets for motorcyclists, AGARD Conference Proceedings 322 Impact Injury Caused by Linear Acceleration, Cologne, Paper 32. Pollitt J.M. (1982), Motorcyclists protective helmets, in Head Protection – The State of the Art, Eds. Pedder J.B. and Mills N.J., University of Birmingham, Birmingham, pp. 55–66. Regulation 22, with amendments 1 to 5, Uniform provisions concerning the approval of protective helmets and of their visors for drivers and passengers of motor cycles and mopeds, United Nations, Geneva. Richter M., Otte D. et al. (2001) Head injury mechanisms in helmetprotected motorcyclists, J. Trauma 51, 949–958. Shuaeib F.M., Hamouda A.M.S. et al. (2002) Motorcycle helmet, Part I. Biomechanics and computational issues, Part II. Materials and design issues, Part III. Manufacturing issues. J. Mat. Proc. Tech. 123, 406–439. Snell Memorial Foundation (2005) Standard for Protective Headgear for Use with Motorcycles, Snell Memorial Foundation, North Highlands, CA. Thom D.R., Hurt H.H. et al. (1997) Feasibility study of upgrading FMVSS No. 218. Motorcycle Helmets, Report to US Department of Transportation. Contract DTNH22-97-P-02001. Vallee H. et al. (1984) The fracturing of helmet shells, IRCOBI Conference, Delft, pp. 99–109. Willinger R. et al. (2000) Dynamic characterization of motorcycle helmets, J. Sound Vib. 235, 611–625. Yettram A.L., Godfrey N.P.M. & Chinn B.P. (1994) Materials for motorcycle crash helmets – a finite element parametric study, Plast. Rubber Comp. Proc. Appl. 22, 215–221. Zellmer H. (1993) Investigation of the performance of motorcycle helmets under impact conditions, SAE Trans. 102(Part 6), 2525–2534.

Chapter 17

Hip protector case study

Chapter contents 17.1 Introduction 17.2 Types of hip protector 17.3 Biomechanics of falls 17.4 Protector test rigs 17.5 Experimental results 17.6 Discussion References

404 405 407 410 416 421 422


Figure 17.1

17.1

Surface appearance and X-ray of a fractured hip (Okuizumi et al., 1998 © American Society of Bone and Mineral Research).

Introduction This chapter discusses a biomedical area where foam products can reduce injuries. It links protector design to the biomechanics of soft tissue deformation and bone fracture. Falls on the hip are not as well understood as head impacts, because few have been reconstructed forensically. A number of test rigs have been designed to rank the products, but there is no general agreement over test methods. Consequently, the test types will be analysed. If the relative effectiveness of products could be established after several years of use, a product standard could be developed. About 240,000 hip fractures occur annually worldwide (Cooper et al., 1992). Hip fractures tend to occur across the neck of the femoral head (Fig. 17.1). The structure of trabecular bone (Fig. 17.2) in the core of the femur is similar to open-cell foams. The spaces are filled with blood cells, marrow, etc. rather than air, but these do not contribute to mechanical strength. The majority of the strength of these long bones comes from the contribution of the compact bone in the outer layers. Figure 17.2b shows that the compact bone in elderly females can also become more porous. Consequently a low bone mineral density (BMD) is a risk factor for hip fractures; the femoral fracture force increases almost linearly with BMD (Etheridge et al., 2005). Sowden et al. (1996) reviewed methods of reducing the number of hip fractures. Osteoporosis is a major factor; hormone replacement

Chapter 17 Hip protector case study 1 mm

700 ␮m (a)

405

Endosteum

Periosteum (b)

Figure 17.2

The structure of (a) trabecular bone reconstructed from CT scan (courtesy of Dr Julian Jones & G.Poolaogasunarmpillai,Imperial College,London) and (b) section of cortical bone between the outer (periosteum) and inner (endosteum) protective membranes (Bousson et al., 2004 © American Society of Bone and Mineral Research).

therapy or diet supplements may be effective in halting further bone density decline, but they cannot reverse the process. Bone strength depends on its condition; exercise encourages the remodelling process by which bone cells (osteons) are replaced and assists the maintenance of balance. A Canadian site www.research.sunnybrook. on.ca covers falls and mobility. Therefore hip protectors are only a small part of the picture. Lauritzen et al. (1993) showed that hip fractures in a particular group of at-risk elderly patients could be reduced by 50% if they wore a particular design of protector. Their results have been confirmed by a trial in nursing homes in Sweden (Ekman et al., 1997). Compliance (the percentage of people wearing the hip protectors) is a major issue. Compliance rates for some products fall significantly with time (van Schoor et al. 2002). Alternatives have been proposed. If the people at risk are confined to a limited area in a residential home it is possible to use energy-absorbing flooring (Badre-Alam et al., 1994; Laing et al., 2006).

17.2

Types of hip protector Protectors should be light in weight, unobtrusive, durable, and cheap. They should stay in place over the trochanter during walking. The articulation of the skeleton and the deformation of the muscles and fatty tissue surrounding the femur and pelvis mean that hip protector designs differ from helmets. Products available in Denmark and Australia are shown on web pages linked to www.hindo.suite.dk.


Figure 17.3

Hip protectors: the top row has foams with a thermoplastic shell and the bottom row, just foams (van Schoor et al., 2006 © Elsevier).

These have been developed from prototypes developed by various medics – Sellberg et al. (1992), Wallace et al. (1993), and Lauritzen et al. (1993). Figure 17.3 shows two types of hip protectors. The first type contains foam pads, while the second type has a thermoplastic shell over the foam layer. The prototype protector used by Lauritzen et al. (1993) had a mass of 45 g. The 2.6 mm thick shell was thermoformed from a glassy thermoplastic and had radii of curvature in the horizontal and vertical directions of 68 and 240 mm, respectively. It was covered on the inside with a 4.5 mm layer of low-density polyethylene (LDPE) foam, of density 64 kg m⫺3, and on the outside there was a 29 mm wide ring of 2.9 mm thick LDPE foam of density 147 kg m⫺3. Its edges fit onto the surface of the thigh, with a central 5 mm gap between the foam and the skin, if the thigh radius is 90 mm. The Danish Safehip product was redesigned (centre of top row of Fig. 17.3) from this prototype. According to US patent 5722093, the prototype was expensive to manufacture and the adhesive bonds between the two foams could not survive washing machine treatments. The shell became a polypropylene (PP) injection moulding with integral holes. A 7 mm thickness of expanded polypropylene (EPP) foam was moulded around this, so the shell is at the mid-thickness rather than the outside of the protector. The EPP foam has a higher compressive yield stress than the foam in the prototype. Two protectors are sewn into pockets in elasticated underwear to keep them in place. There has been debate (see US patent 5599290) on whether hip protectors should shunt impact forces away from the head of the femur, as well as attenuating directly transmitted forces. Six energy

Chapter 17 Hip protector case study

407

B E C A

F

D

B Floor Flesh

Figure 17.4

Shell Foam

Mechanisms of energy absorption of a hip protector with a shell (see text for key).

absorption mechanisms are possible when a protected thigh hits the ground (Fig. 17.4): (A) (B) (C) (D) (E)

compression of the flesh directly over the greater trochanter; loading of the surrounding muscles and fat; energy absorption in the hip joint; energy absorbed by the floor or clothing; flexing of the shell of the protector, especially if there is an initial gap between the protector and the flesh over the trochanter; (F) compression of the foam in the protector. If a protector is not worn only the first four mechanisms act, and in the foam pad type mechanism (E) is missing and (B) is minor.

17.3

Biomechanics of falls 17.3.1 Factors affecting hip fracture Cummings and Nevitt (1989) proposed four necessary conditions for hip fracture: 1. 2. 3. 4.

a fall causes a hip impact; protective responses, such as breaking the fall with arms, fail; the local soft tissue absorbs insufficient energy; the residual fall energy exceeds that needed to fracture the proximal femur.

The soft tissue thickness over the greater trochanter ranges, for healthy women aged more than 65 years, from 20 to 60 mm, with

408 Polymer Foams Handbook the majority in the 20–40 mm range (Maitland et al., 1993). Flesh is viscoelastic (so loading rate affects the response) and non-linear (Chapter 5). Therefore its properties should ideally be determined at high strains and impact strain rates. Soft tissue properties change with the age of the person, and the most relevant data are for those at risk. As the thickness of soft tissue over the trochanter varies, measurements would be needed on a number of elderly people. However, ethical permission would not be granted to cause bruising. Robinovitch et al. (1995a) measured the static indentation stiffness of the soft tissue over the trochanter of young adult females; the flat end of a cylinder of radius 38 mm was pressed with a force of 60 N, giving a value of 35 ⫾ 14 N mm⫺1. However, the maximum force is much less than those experienced in falls. Consequently, Robinovitch et al. (1995b) measured the impact response of cadaver trochanteric flesh using the impact rig shown in Figure 17.9a. The displacement is the sum of that of the flesh and the rubber support springs; in the case of the 2 mm skin sample, it is clearly mainly due to the latter. This graph shows how the test rig underestimates impact forces; with the minimal protection 2 mm of skin, the peak force from a 113 J impact is only 6.5 kN. Laing et al. (2006) dropped the hips of young female volunteers onto a force plate; at the maximum drop height of 50 mm, the peak forces approached 1.3 kN. However, they did not compute the force vs. deflection graphs. It was impossible to find data for the compressive properties of fat, muscles, or fascia under impact conditions, loaded 7

2 8

Force on femur (kN)

6 5

4.3 4 3 2 1 0

0

10

20

30

40

50

60

70

Displacement (mm)

Figure 17.5

Force vs. deformation, for various thicknesses (mm) of trochanteric tissue using the rig shown in Figure 17.9a (redrawn from Robinovitch et al., 1995b © Orthopaedic Research Society).


409

to high stresses. It is not acceptable to injure volunteers, while tests on fatty layers from pigs (Hipp et al., 1991) may not be relevant to humans. Hinton and Smith (1993) showed that for elderly white female patients, 49% of femoral fractures were trochanteric, 47% transcervical, and 3% sub-trochanteric. The forces to fracture isolated femurs, loaded on the femoral head while the trochanter was supported, were found to be in the range 0.8–4.0 kN (Lotz and Hayes, 1990) for a slow loading rate (0.7 mm s⫺1). The failure load depended on the fracture site and the age of the femur, and had a mean value of 2.5 kN for inter-trochanteric fractures. Finite element analysis (FEA) showed that the inter-trochanter region experienced the highest stresses in a fall (Lotz et al., 1991). The strength of bone is rate dependent; and the mean strength of femurs, from females aged 73 ⫾ 7 years, was 4.1 ⫾ 1.6 kN at a loading rate of 100 mm s⫺1, 21% higher than in tests at 2 mm s⫺1 (Courtney et al., 1994). The force required to fracture a trochanter depends on the force direction relative to the cervical neck. If the impact force is applied perpendicular to the trochanter surface, the fracture force should be somewhat higher than the values quoted by Lotz et al.

17.3.2 Fall energy Lotz and Hayes (1990) estimated the kinetic energy of falls causing hip fracture by multiplying the trochanteric height of adult females by their body mass, obtaining 470 ⫾ 150 J. However hip impact velocities were measured as 3.2 ⫾ 0.5 m s⫺1, equivalent to a free fall from 0.52 m, rather than from the hip height of 0.8 m (Van der Kroonenberg et al., 1993). Taking the effective mass of the hip as 8 kg, the kinetic energy of the impact is about 40 J at the upper end of the range (5–50 J) to fracture isolated femurs. Askegaard and Lauritzen (1995) used a young stuntwoman, who fell sideways onto a 50 mm thick layer of (rigid polyurethane?) foam, of compressive collapse stress 150 kPa, supported on a loadsensing platform. They moved the platform suddenly to the side, so she fell with her knees unbent. However elderly women’s torsos are likely to fall nearly vertically, as their knees bend. The indentations in the foam showed a maximum hip contact area of about 350 cm2, while the maximum force was 3.5 kN. They reproduced the force–time history of the fall (Fig. 17.6) by dropping a physical model, consisting of two masses connected by a spring, onto the foam. A mass m1 ⫽ 3 kg, having the shape of the stuntwoman’s hip imprint, represented the bone mass in the pelvic region, plus some surrounding tissue. The mass m2 ⫽ 16 kg represented the effective mass of the torso and the spring constant k ⫽ 110 N mm⫺1. The


Impact force (kN)

3

2

Person 1 Mechanical model 0 25

50

75

100

125

Time since trigger pulse (ms)

Figure 17.6

The force–time history of the fall of a stuntwoman’s hip onto a foam layer and that of a mechanical model (redrawn from Askegaard and Lauritzen, 1995 © Scandinavian University Press).

modelling parameters would be different if the falls were onto a less compliant surface.

17.4

Protector test rigs 17.4.1 Choice of a flesh substitute The trochanter should be covered with a material, of similar thickness and mechanical properties to the muscles, fat, and fascia of the hip, which should not deteriorate with use or with time. In vivo, muscle and fascia are free to expand sideways over the surface of the trochanter; so the compressive properties should be directly related to the ease of shearing. When flesh of density 1000 kg m⫺3 is impacted at a high velocity, its apparent stiffness is higher than in a quasi-static test, as its mass must accelerate away from the striker. Figure 17.7 shows a cross-section of a cadaver and the soft tissue cover of the greater trochanter. A variety of materials have been used as flesh substitute: soft rubbers, gels, and foams. Foams are unsuitable since their near-zero Poisson’s ratio allows compression without lateral expansion, and their low density allows acceleration by a low force. However, when hit with a high-energy blow, the foam bottoms out and acts as a rigid layer. van Schoor et al. (2006) used slow recovery polyurethane


Figure 17.7

411

Cross-section of a female cadaver at the level of the hip. Key: 17, greater trochanter; 33, acetabulum; and 34, femoral head (Ellis et al., 1991 © Butterworth Heinemann).

Confor 45 foam of density 100 kg m⫺3, which is used in dummy legs for pedestrian safety testing. This is likely to provide more energy absorption than human flesh. Derler et al. (2005) used a silicone rubber of unspecified Young’s modulus; it is likely be larger than that of human tissue. Mills (1996) used an oil-extended jelly-like red polymer of density 1110 kg m⫺3. Its compressive force–deflection relationship, measured under impact conditions, is highly non-linear (Fig. 17.8). The response becomes very non-linear as the deflection approaches the flesh substitute thickness. The stiffness for loading to a force of 600 N is approximately 70 N mm⫺1. This is twice Robinovitch’s static value (1995a) for human subjects. At a low strain rate of 0.02 s⫺1, Young’s modulus was found to be 0.23 MPa for strains up to 20% and with a non-linear response at higher strains. Repeated impacts show that its response is constant for at least 10 impacts. It was cast around a replica of the hip and load cell with a wooden box constraining the extremities of block. A 20 mm thickness was used over the trochanter to simulate people most at risk of hip


Force (kN)

6

4

3

2 2 1

0

0

5

10

15

20

25

Distance (mm)

Figure 17.8

Impact force vs. deflection for 20 mm of flesh substitute over the trochanter for impacts from the marked heights (m) (Mills, 1996).

fracture. The outer surface was cylindrical with a ‘horizontal’ radius of curvature of 90–100 mm, matching that from a small survey of people. Its total mass was 2.46 kg, and the upper surface was covered with a thin layer of talc-coated surgical rubber glove to act as a skin. The layer of flesh substitute was inspected after every 10 tests and replaced if necessary. Damage occurred, particularly when the forces on the trochanter exceeded 6 kN.

17.4.2 Effective mass of the thigh Given the large number of test rigs that exist, examples of the two main kinds will be discussed. The first uses a typical lower torso mass with rubber springs behind the ‘hip’ to limit the impact forces. In the second kind the head of the femur is rigidly supported. In all the test rigs a large rigid mass with a flat face (the ‘floor’) is moved with one degree of freedom (along one axis or rotating about a single axis) while the support of the ‘hip’ is fixed (Fig. 17.9). This is more convenient than dropping an instrumented hip onto a fixed floor. As the protector mass is negligible compared with the falling mass, no errors are introduced by having a fixed hip. The impact is direct, perpendicular to the surface of the greater trochanter. Consequently the stability of the protector positioning is not tested, as it would be during a slightly oblique impact.


413

1.7 m

Femoral load cell

Polyethylene foam soft tissue covering

Pendulum mass Load cell

Femur

Neoprene pelvic bumper spring

(a)

5 0 −5 −10 −15 −20

0

20

40

(b)

Figure 17.9

Test rigs of: (a) Robinovitch et al. (1995a © ASME) and (b) Derler et al. (2005 © Elsevier).

The effective hip mass Me produces the same peak force on the trochanter as does a human falling with the same velocity. Me is greater than the mass of the pelvis and upper parts of the femur of circa 0.9 kg. Viano (1989) measured the peak force F (N) for pelvic impacts on cadavers at a range of impact velocities V (m s⫺1). When F was plotted against V, the line of best fit was F ⫽ 1230V with a correlation coefficient of 0.85. The analysis of the impact between two masses, with a contact stiffness k (Mills and Zhang, 1989), gives the height of the initial force peak as I =

∫

wy2 dy

(17.1)

For a 20 mm thickness of a flesh substitute over a trochanter (Fig. 17.8) k, the slope of the graph is 178 N mm⫺1; hence the effective mass needed for Mills (1996) test rig was 8.5 kg.

414 Polymer Foams Handbook V

Striker m1

n1

k1

Femur mass m2

n2

k2

Fixed table

Figure 17.10 Lumped-mass model for the Robinovitch et al. (1995a) test rig (Mills, unpublished).

17.4.3 Modelling of test rig designs The response of Robinovitch et al. (1995a) test rig (Fig. 17.9a) can be analysed using lumped-mass models explained in Section 16.7 (Fig. 17.10). They used the elastic version of this approach in developing their test rig: a striker mass of M1 ⫽ 35 kg, representing the torso, hit a surrogate femur of unspecified mass M2, supported on a rubber springs having total spring constant 70 N mm⫺1. A 13 mm thick pad of polyethylene foam (of density 32 kg m⫺3 according to Badre-Alam et al. (1994)) simulated the tissue over the trochanter. Its indentation stiffness of 35 ⫾ 14 N mm⫺1 matched that measured statically on young adult females. The rig response for low-energy impacts, without a protector, onto the foam was found to be the same for impacts onto cadaveric tissue. The striker mass had an impact velocity of 2.6 m s⫺1, so 118 J kinetic energy. Without a protector, the peak force was 6.5 kN, similar to the skin response in Figure 17.5. Hence the test does not reveal the inadequacy of poor protectors. The kinetic energy is too severe for designs that are effective in wearer trials. Nevertheless their tests indicated that their design was the best of eight products tested! Modelling shows that the main rubber springs are non-linear. If they were linear the shape of the force vs. time t response would be close to A sin t for π ⬎ t ⬎ 0; the typical non-linear hardening of a rubber block in compression modifies the shape to approximately an inverted V (Fig. 17.11a). The rubber springs are estimated to deflect

Chapter 17 Hip protector case study 6

6

5

No pad

4

Impact force (kN)

Impact force (kN)

5

Pad D

3 Pad H

2 1

4 3 2 1

0

0 0

(a)

415

20

40 60 Time (ms)

80

100 (b)

0

20

40

60

80

100

Time (ms)

Figure 17.11 (a) Striker (dashed line) and femur force (solid curve) vs. time (Robinovitch et al.,

1995a © ASME). (b) Simulation using the parameters k1 ⫽ 35, k2 ⫽ 70 kN m⫺1, n1 ⫽ 1000, n2 ⫽ 100 Ns m⫺1, m1 ⫽ 35, m2 ⫽ 1.5 kg, drop height 0.35 m (Mills, unpublished).

by 35 mm and then return about 60 J energy on rebound. A response similar to that of pad D (Fig. 17.11a) can be predicted by the lumped mass model (Fig. 17.11b) showing that the pad acts as a damped spring. Pad D is a prototype by Wortberg, while pad H is Robinovitch et al.’s ‘energy shunting pad’. Parkkari et al. (1994) used a related rig with a hip mass M2 ⫽ 11.3 kg, supported on elastic steel springs of k ⫽ 75 N mm⫺1. As there was no soft tissue, the hip protector shell cannot divert load from the femoral head. The mass M2 on steel springs is prone to resonate; modelling shows that the striker force can have multiple peaks, due to its series of impacts with the resonating mass M2. This faulty test rig design cannot evaluate the benefit of energy shunting protector designs. Mills’ (1996) test rig rigidly supported the ‘femur’ because of the design faults of the above-mentioned rigs. The upper femur was machined from a 42 mm diameter rod of ABS (acrylonitrile butadiene styrene) copolymer. The nearly flat area, of length 40 mm and width 20 mm, towards the rear of the trochanter was used as the impact site, rather than the small radius protuberance on the trochanter. There is negligible energy absorbed by the ABS, and there is little striker rebound. A quartz piezoelectric load cell, between the trochanter and the concrete floor, measured the force on the trochanter. The striker force was calculated from the product of its mass and its linear acceleration. Its position was calculated from its acceleration and pre-impact velocity in the same manner as for cushion curve tests. The striker surface was covered with a 6 mm thickness of foam-backed carpet to represent a typical floor. The rigs of

416 Polymer Foams Handbook Derler et al. (2005) shown in Figure 17.9b and van Schoor et al. (2006) are similar, but constructed to a higher precision, using steel for the femur.

17.4.4 Test criteria Hip protector test criteria should be conservative until field trials have established the product effectiveness. Therefore a force limit of 2.5 kN was used by Mills (1996). Other researchers chose to measure the peak force for an impact of a specific energy and used this as a kind of ranking.

17.5

Experimental results 17.5.1 Ranking of protectors Derler et al. (2005) did not reveal the identity of the 10 types of hip protector tested, but they divided them into those with and without shells. The 10 kg mass falling through 0.5 m (50 J impact energy) caused peak femoral forces in the range from 2 to 4 kN, depending on the protector (Fig. 17.12a). van Schoor et al. (2006), using 12.7 thickness of Confor foam as a flesh stimulant, found that hip protectors with shells were superior to those without (Fig. 17.12b). The 25 kg mass falling through 0.08 m (20 J impact energy) produced forces that should not cause a femoral fracture if a protector with a shell was worn.

17.5.2 The striker force–deflection relationship To allow the development of improved protectors, it is important to generate data that relate to the materials used and the product dimensions. However, most researchers report the peak forces in impacts and do not consider these issues. The only other force F vs. deflection x data that could be found was a single graph in Derler and Spierings (2004). Figure 17.13 shows typical graphs of striker force vs. the deflection of the upper surface of the hip protector (Mills, 1996). Loading traces for different drop heights approximately superimpose on a master curve, so the strain rate dependence of the flesh substitute and protector foams is moderate. The flesh substitute deforms between the trochanter, with a double convex curvature, and the inner surface of the protector shell, which becomes flat in its central area when in contact with the striker surface. The strain distribution in the flesh substitute is non-uniform,


417

Peak force (N) 8000 I

G

E

F

J

C

B

A

D

H

7000 6000 5000

Maximum force (kN)

5 4

Fimpact Fneck

4000 3000

3 2000 2

1000

1 1

0 I

G

E

F

C

B

A

D

3

4

5

6

7

8

9 10 11 12

Hard HP

H

Soft HP P < 0.001

(b)

Hip protector

(a)

Figure 17.12

J

2

Product comparisons: (a) grey bars: peak external force, black bars: transmitted femoral force (Derler et al., 2005 © Elsevier) and (b) peak femoral force with error bars 1: bare femur, 2: femur plus confor foam (van Schoor et al., 2006 © Elsevier). 6

6

Force (kN)

4

5 3

2

0

0

10

20

30

40

50

Distance (mm)

Figure 17.13 Striker force vs. deflection for a Sahva protector over 20 mm of flesh substitute on the trochanter of the test rig for impacts from 0.2, 0.3, and 0.6 m (Mills, 1996).

with the highest strains at the centre of the trochanter. Its total thickness over the trochanter was either 10 or 20 mm. Figure 17.8 shows that force only rises above 5 kN if the flesh substitute is compressed to less than 5 mm thick. Therefore the high striker force is mainly


Impact energy to produce a peak trochanter force of 2.5 kN Flesh thickness (mm) Flesh radius (mm) Protector 10 20 10 20 20 20

140 100 90 100 100 100

Impact energy (J)

None None Sahva Sahva Shell ⫹ 15 mm LD70 Shell ⫹ 15 mm PP30

8 16 19 33 57 66

directed, via the highly compressed flesh substitute, to the greater trochanter. The impact energy (calculated as an integral of the force–deflection graph) at which the trochanter force was 2.5 kN was used as the protector performance measure (Table 17.1). When the thickness of the flesh substitute was increased from 10 to 20 mm, the energy absorbed increased in proportion. Since the 6 mm carpet alone absorbed 3 J, the 20 mm of flesh over the trochanter absorbed 13 J. The Sahva prototype protector increased the ‘safe’ impact energy (for 20 mm flesh cover) by 17 J. When the Sahva protector shell was used with 15 mm thicknesses of PP or LDPE foams, their protective capacity was 40–50 J higher. Hence, foam that is 10 mm thicker than that in the Sahva prototype (4.5 mm), with a higher compressive resistance, offers a higher impact protection level.

17.5.3 Load sharing between the femoral head and surrounding flesh In a rig where the striker acceleration aS and the force on the fixed femoral head FFH are measured, the force on the flesh can be computed using Fflesh ⫽ mS aS ⫺ FFH

(17.2)

where mS is the striker mass. Figure 17.14 compares the force–time traces with and without the Sahva prototype protector. The force on the flesh rises slightly earlier than that on the trochanter, as the flesh accelerates away from the striker. The Sahva protector (Fig. 17.14b) reduces the peak trochanter force by 63% while a 1.2 kN force is distributed to the surrounding flesh, and the impact is spread over a longer time. The impact impulse was defined in Chapter 14. The protector increased the impulse transmitted to the flesh (Table 17.2) from 12% to 32% of the total. For minor falls from 0.2 m, the Sahva prototype

5

2.5

4

2 Force (kN)

Force (kN)


3 2

(a)

1.5 1 0.5

1 0 10

419

20

30 Time (ms)

40

0 10

50

20

(b)

30 Time (ms)

40

50

Figure 17.14 Forces on the trochanter (thin line) and the surrounding flesh (thick line) for 0.2 m impact on the trochanter covered with 20 mm of flesh substitute: (a) without and (b) with the Sahva protector (Mills, 1996).

Table 17.2

Impulse on the flesh and on the hip (20 mm of flesh substitute)

Protector None Sahva Sahva Shell ⫹ 15 mm LD70 Shell ⫹ 15 mm PP30

Drop height (m)

Impulse on hip (Ns)

Impulse on flesh (Ns)

Peak hip force (kN)

Peak flesh force (kN)

0.2 0.2 0.4 0.5 0.5

28.7 20.3 32.8 30.8 30.5

3.9 19.3 15.8 20.8 23.8

3.91 1.67 4.53 2.85 2.12

0.63 1.20 1.52 1.60 1.53

LD70 is LDPE foam and PP30 is PP foam (see Table 17.3).

protector distributes around half of the impact force away from the trochanter, but for a 0.4 m fall the trochanter receives the majority of the impact force. Once the fascia and muscle above the trochanter is subjected to a high compressive strain, the direct load path from the hip to the floor causes high forces on the proximal femur. It is not possible to calculate the energy absorbed in the surrounding flesh, since the movements of the shell and flesh surfaces are unknown. The better performance with a 15 mm thickness of PP foam is due to energy absorption in the foam and possibly a greater deflection of the protector shell. The initial central gap between the inner surface of the Sahva protector and the outer surface of the flesh substitute was 5 mm. The gap varies with the radius of the wearer’s leg and with the pressure exerted by clothing. When the gap was increased to 11 mm, there was little extra impact protection.



0.5

0.4

0.3

0.2 Energy density input for σ < 0.3 MPa

0.1

0 0

20

40

60

80

Strain (%)

Figure 17.15 Compressive stress–strain curves under impact conditions for PP30 (solid line) and LDPE70 foams (dashed line).The area under the graph, to the left of the dotted line, represents the energy density input for stress ⬍0.3 MPa (Mills, 1996).

17.5.4 Foam selection The foam type and thickness should optimise the energy absorption. A foam thickness limit of 15 mm was set for the product to be wearable. A foam, of higher yield stress than the 64 kg m⫺3 density LDPE foam in the Sahva prototype, could be positioned directly above the trochanter. If the head of the trochanter can tolerate a force of 2.5 kN and its surface area is 20 mm by 50 mm, the average compressive stress can be up to 2.5 MPa. If the foam is stiffer than the flesh of the thigh, the flesh will be pushed away from the foam, without significant energy absorption in the foam. The curved area of the inside of the Sahva shell was measured as 10,700 mm2. If the protector distributes 3 kN force to the flesh surrounding the trochanter, the average compressive stress on the foam is 0.28 MPa. Therefore the foam should absorb as much energy as possible without the stress exceeding 0.3 MPa. Foam impact stress–strain curves were measured at room temperature (Fig. 17.15). Table 17.3 gives the areas under the stress–strain curves (energy input per unit volume of foam) at the 0.3 MPa stress level. The compressive properties of the particular LDPE and PP foams in Table 17.3 are better than the soft ethylene–vinyl acetate (EVA) foam. The energy absorbed, by a 15 mm thickness and 150 cm2 area of foam when the stress is 0.3 MPa, is given in the last column of Table 17.3. By comparison with test results in Table 17.1, roughly this amount of foam appears to be deformed in the tests.


Table 17.3

421

Impact compressive data on foams at 20°C

Foam LDPE PP EVA

Density (kg m⫺3) 70 30 35

Energy density J/m at stress 0.3 MPa 0.11 0.12 0.07

Energy J at 15 mm thick and 150 cm2 area 26 28 17

17.5.5 Shell stiffness evaluation The protector shell should distribute the impact force to a large area of the underlying foam, and hence to the thigh soft tissue. Its effectiveness in doing this depends on its bending stiffness when loaded centrally. Since glassy thermoplastic shells are relatively rate independent, the bending stiffness of Lauritzen’s prototype shell was measured at a low loading rate, by compressing it between two flat parallel plates. The slope of the force–deflection curve (Fig. 17.16) for the first 5 mm of loading is only 21 N mm⫺1, so a negligible energy of 0.25 J is stored elastically. The slope then rises, but at a force of 1.1 kN the shell buckles inwards, causing the force to fall. At a deflection of 22 mm, the shell is nearly flat and the energy input is 14.8 J. The internal PP shell of the ‘Safehip’ protector stiffens in a nonlinear manner when a central force attempts to flatten it. When the hip protector is tested on a surrogate hip, external forces are applied to the centre of the shell, with internal support over much of the inner surface. Hence the shell deflection at a given striker force will be somewhat less than that shown in Figure 17.16. Schmitt et al. (2004) used dynamic FEA to model a hip protector on a test rig. However there was no detail of the deformation of the protector shell or the load spreading to the silicone rubber flesh substitute; their only concern was to replicate the force–time traces of a typical impact.

17.6

Discussion Hip protectors should ideally divert much of the impact force to the muscle and fatty tissue surrounding the trochanter. If the form of the protector force–deflection graph could be made convex downwards, more energy could be absorbed safely. Some hip protector assessments of the 1990s were biased towards the ‘home’ prototype product. Product ranking depends on the rig design and test pass/fail criterion, as well as ergonomic factors. A ranking using an inappropriate rig is unlikely to correspond with the performance in practice. Ten years later, there are many products


Force (kN)

1.5 Buckles 1.0

Unbuckles

0.5

0.0

0

5

10

15

20

25

Displacement (mm)

Figure 17.16 Force–deflection graph for the slow loading and unloading of the Sahva prototype shell between two flat steel plates (Mills, 1996).

on the market, and large-scale wearing trials have been carried out in several countries. There have been epidemiological surveys of effectiveness (Parker et al., 2006). Forensic reconstructions of falls could establish protection levels. However, the foams used in most hip protectors fully recover their dimensions after impact. Therefore visual inspection cannot reveal the impact site, and there is no published analysis of protectors which have worked. Protector designs are a compromise between protection level and wearability. Thicker foams of greater protective capacity could be used and stiffer shells employed, but this would make the device bulkier. Further research is needed to measure the pressure distribution on the upper leg in a fall and the properties of the impacted tissues under impact conditions. Such information is needed for the optimisation of protector design. In future FEA will be used, with better data for the soft tissue properties, to simulate the load shunting properties of protectors In general this is a less well-defined area than helmets. The population to be protected varies considerably in soft tissue cover, BMD, weight, and height.

References Askegaard V. & Lauritzen J.B. (1995) Load on the hip in a stiff sideways fall, Eur. J. Muscoloskel. Res. 4, 111–115.


423

Badre-Alam A., Casalena J.A. et al. (1994) A dual stiffness floor for the reduction of fall injuries, Proceedings of the 4th CDC Symposium on Injury Prevention Through Biomechanics, Wayne State University Detroit, MI, pp. 11–19. Bousson V., Peyrin F. et al. (2004) Cortical bone in the human femoral neck: 3D appearance and porosity using synchrotron radiation, J. Bone Min. Res. 19, 794–801. Cooper C., Campion G. & Melton L.J. (1992) Hip fractures in the elderly: a world wide projection, Osteoporosis Int. 2, 285–289. Courtney A.C., Wachtel E.F. et al. (1994) Effects of loading rate on strength of the proximal femur, Calcif. Tissue Int. 55, 53–58. Cummings S.R. & Nevitt M.C. (1989) A hypothesis: the causes of hip fracture, J. Gerontol. 44, M107–M111. Derler S. & Spierings A. (2004) Wirksamkeit von Huftprotektoren, EMPA-Bericht No. 262, St. Gallen, Switzerland. Derler S., Spierings A.B. & Schmitt K.U. (2005) Anatomical hip model for the mechanical testing of hip protectors, Med. Eng. Phys. 27, 475–485. Ekman A., Mallmin H. et al. (1997) External hip protectors to prevent osteoporotic hip fractures, Lancet 350, 563–564. Ellis H., Logan B. & Dixon A. (1991) Human Cross-Sectional Anatomy, Butterworth Heinemann, London. Etheridge B.S., Beason D.P. et al. (2005) Effects of trochanteric soft tissue and bone density on fracture of the female pelvis in experimental side impacts, Ann. Biomed. Eng. 33, 248–254. Hinton R.Y. & Smith G.D. (1993) The association of age, race and sex with the location of proximal femur fractures in the elderly, J. Bone Joint Surg. 75A, 752–759. Hipp J.A., Myers E.R. et al. (1991) Soft tissue and energy absorption capacity as potential determinants of hip fracture risk, Transactions of the 37th Orthopaedic Research Society, Vol. 16, p. 435. Laing A.C., Tootoonchi I. et al. (2006) Effect of compliant flooring on impact force during falls on the hip, J. Orthop. Res. 24, 1405–1411. Lauritzen J.B., Peterson M.M. & Lund B. (1993) Effect of external hip protectors on hip fractures, Lancet 341, 11–13. Lotz J.C. & Hayes W.C. (1990) The use of qualitative computed tomography to estimate the risk of fracture of the hip from falls, J. Bone Joint Surg. 72A, 689–700. Lotz J.C., Cheal E.J. & Hayes W.C. (1991) Fracture prediction for the proximal femur using the finite element method, Part 1, J. Biomech. Eng. 113, 353–360. Maitland L.A., Myers E.R. et al. (1993) Read my hips: measuring trochanteric soft tissue thickness, Calcif. Tissue Int. 52, 82–87. Mills N.J. (1996) The biomechanics of hip protectors, Proc. Inst. Mech. Eng. Part H J. Eng. Med. 210, 259–266.

424 Polymer Foams Handbook Mills N.J. & Zhang P.S. (1989) The effects of contact conditions on impact tests on plastics, J. Mater. Sci. 24, 2099–2109. Okuizumi H., Harada A. et al. (1998) Effect on the femur of a new hip fracture preventative system using dropped-weight impact testing, J. Bone Min. Res. 13, 1940–1945. Parker M.J., Gillespie W.J. & Gillespie L.D. (2006) Effectiveness of hip protectors for preventing hip fractures in the elderly people: systematic review, BMJ online. Parkkari J., Kannus P. et al. (1994) Force attenuation properties of various trochanteric padding materials, J. Bone Min. Res. 9, 1391–1396. Robinovitch S.N., Myers, E.R. et al. (1994) Hip fracture protection from falls in the elderly, Proceedings of the CDC Symposium on Injury Prevention Through Biomechanics, Wayne State University Detroit, MI. Robinovitch S.N., McMahon T.A. & Hayes W.C. (1995a) Energy shunting hip padding system improves femoral impact force attenuation in a simulated fall, J. Biomech. Eng. 117, 409–413. Robinovitch S.N., McMahon T.A. & Hayes W.C. (1995b) Force attenuation in trochanteric soft tissues during impact from a fall, J. Orthop Res. 13, 956–962. Schmitt K.U., Spierings A.B. & Derler S. (2004) A finite element approach and experiments to assess the effectiveness of hip protectors, Tech. Health Care 12, 43–49. Sellberg M.S., Huston J.C. & Kruger D.H. (1992) Development of a passive protective device for the elderly to prevent hip fracture, ASME Adv. Bioeng. 22, 505–508. Sowden A., Sheldon T. et al. (1996) Preventing falls and subsequent injuries in older people, Effect. Health Care 2(4). NHS Centre for Reviews and Dissemination, University of York, pp. 1–16. Van der Kroonenberg A., Munih P. et al. (1993) Hip impact velocities and body configurations for experimental falls from standing height, Transactions of the 39th Annual Meeting of Orthopaedic Research Society, Vol. 18, p. 24. van Schoor N.M., Deville W.L. et al. (2002) Acceptance and compliance with external hip protectors, Osteoporosis Int. 13, 917–924. van Schoor N.M., van der Veen A.J. et al. (2006) Biomechanical comparison of hard and soft hip protectors, and the influence of soft tissue, Bone 39, 401–407. Viano D.C. (1989) Biomedical responses and injuries in blunt lateral impacts, Proceedings of the 33rd Stapp Car Crash Conference, pp. 113–142. Wallace R.B., Ross J.E. et al. (1993) Iowa FICSIT Trial, The feasibility of elderly wearing a hip joint protective garment to reduce hip fractures, J. Am. Geriatr. Soc. 41, 338–340.

Chapter 18

Sandwich panel case study

Chapter contents 18.1 Introduction 18.2 Sandwich panel types 18.3 Elastic design of sandwich panels 18.4 Failure modes 18.5 Sport applications 18.6 Thermal insulation panels Summary References

426 426 431 436 440 440 445 445


18.1

Introduction This chapter analyses the properties behind sandwich panel applications in the construction, transport, and sport fields. It emphasises the foam properties needed to compliment the properties of the surface skin materials. Panels for boat building require high compressive strengths and impact toughness, so specialised polyvinyl chloride (PVC) foams of relatively high density are used. The main role of panels for house and industrial buildings is thermal insulation, so the foams are of lower density. The thermal conductivity of foams is considered later in the chapter. The insulating foams in the walls of domestic freezers or refrigerators have thermoplastic skins. The inner facing is in contact with the food, while the outer skin is hidden by the metal casing. There is a large literature on the mechanics of sandwich beams; only the basic bending and torsional theory are explained here, plus aspects of foam shear which relate to the other parts of the book. Sandwich beam research is isolated from other types of foam research. Some findings of papers analysing sandwich panel performance are relevant to case studies in this book. In general, sandwich panels have two flat skins, which are bonded to the foam core. In contrast, sport protectors, helmets, and hip protectors have an outer shell with single or double curvature, which is generally not bonded to the foam (the exception being some bicycle helmet microshells). Double curvature provides much greater resistance to buckling under surface loads than flat skins.

18.2

Sandwich panel types 18.2.1 Sandwich panels for buildings In Europe Sandwich Panels, or in the USA Structural Insulation Panels (SIP), consist of rigid foams faced with metal or wooden skins. Koschade (2002) and Zenkert (1997) have reviewed the subject. Hartsock (1991) pointed out that foam-filled panels provide excellent thermal insulation, can carry significant loads, while being easy to erect. SIPs are used widely for the construction of house walls and roofs. The wood veneer skins, bonded to the foam core, produce a sandwich panel with greatly increased structural stiffness and strength compared with plywood. Rival products use honeycomb cores, with the cell axis perpendicular to the skin. The thermal conductivity of these foams, their resistance to water penetration, and their mechanical properties are emphasised. The optimum density for thermal resistance is lower than that for mechanical properties, so if both are required, the density is a compromise. Sandwich panels can have smooth skins (Figure 18.1) or skins with

Chapter 18 Sandwich panel case study

Figure 18.1

427

A 47 mm thick sandwich insulation panel with EPS core and hardboard skins.

longitudinal ridges for increased buckling resistance in one direction. In Australia such panels can contain polystyrene (PS) foam; however this means that the ridged sections of the steel skin are empty. Flat skins could be adhesively bonded to sheets of already-manufactured foam. When sandwich panels replace conventional brick or concrete-block walls, the construction technology changes. The panels are constructed off-site; Koschade (2002) describes the uncoiling and possible profiling of aluminium skins before the polyurethane (PU) chemicals are applied to the centre of the sandwich. The ridged or curved aluminium skins provide the ‘mould’ for the initially liquid chemical for PU foam. The expanding foam fills the profile then, as it polymerises and cross-links, provides adhesive bonding. The foam supports the thin skins, preventing local buckling, making the sandwich panel mechanically efficient. It is this synergy that makes the systems so attractive. The panels are cut into lengths up to 8 m and then delivered to site. They are significantly thinner than brick plus cement-block cavity wall with cavity insulation. In a typical 100 m by 100 m building, another 10% of the total area is useful space rather than wall. A steel girder framework can be erected quickly; it does not need to have diagonal bracing members, as the bolted-on sandwich panels provide the shear resistance. This means that the shear and torsional stiffness of the panels are equally important as the bending stiffness. The details of jointing, rainwater paths, and thermal expansion joints differ from those in conventional buildings. Koschade (2002) quotes data for rigid PU foams showing that the tensile Young’s modulus exceeds the flexural Young’s modulus, which in turn exceeds the compressive Young’s modulus. This suggests faults in the test method (Chapter 5) and/or foam inhomogeneity and anisotropy.

428 Polymer Foams Handbook Table 18.1 Ranking of materials for sandwich cores for boat construction (1 is best) Property

Density (kg m⫺3)

Weight factor Divinycell HD 90–250 Aluminium 41 honeycomb Balsa wood 96 PP honeycomb 78 Hexcel NP 72

Mechanical properties

Corrosion resistance

Fire resistance Repairability Cost Joining

2 1 4

3 4 3

2 2 3

2 3

1 1 2

2 2 2

27 33

2 1 3

4 4 2

1 3 2

1

4 4 3

2 2 2

31 32 26

Total

18.2.2 Sandwich panel cores for boat building Jackson et al. (1999) listed rival core materials (Table 18.1) for the construction of lightweight, fast ships. The weighting factors in the first row were used to calculate the total score in the last column. The Divinycell foam has the same score as the Hexcel honeycomb made from aramidfibre in phenolic resin. They discuss problem areas such as fire resistance, joining the panels, the effects of impacts, and repair methods. Glass-fibre reinforced polyester resin is the skin material of choice for boat building. It is not possible to use PS foam cores, as the resin dissolves the foam. Consequently specialised closed-cell PVC foams, with increased toughness, are used. They have low absorption of water or polyester resin and no chemical reactions with the latter. Various types are used: 1. Linear PVC foam, made by Airex. 2. Cross-linked PVC foam, made by Herex. 3. PVC interpenetrating polymer network, made by Diab. Danielsson (1996) described the use of Divinycell HD foam from Diab, which consists of interpenetrating polymer networks of PVC and a cross-linked polyurea/polyamide system. The foam is made by a threestep process. First, a plastisol is made from a mixture of PVC powder, anhydrides, isocyanates, and blowing agents. Second, the plastisol is gelled under pressure in a mould at an elevated temperature, during which some bubble nucleation occurs. Third, when the gel is heated in water at between 80°C and 100°C, the CO2 evolved reduces the foam density to as low as 30 kg m⫺3. Amines, from the water– isocyanate reaction, react with anhydrides and isocyanates to form a network of polyamide, polyimide, and polyurea that interpenetrates the PVC. The foam is post-cured in water at 70°C for 2–25 days, to consume the residual isocyanates. Figure 18.2 shows the dynamic Young’s modulus and tan δ, as a function of temperature. The main transition, at about 70°C, is due to the Tg of the PVC.


429

0.3

1 3 4 2

8

0.2 Tan δ

Log (Young’s modulus E ) (Pa)

9

7 2

1

0.1

3 4

6

0 5

−100

−50

0

50

100

Temperature (°C)

Figure 18.2

Dynamic mechanical thermal analysis for Diab foams of density (1) 89, (2) 121, (3) 164 and (4) 221 kg m⫺3 measured in bending (Danielsson, 1996 © RAPRA).

4.5

Peak stress (MPa)

4.0 3.5 3.0 H 130 HD 130

2.5 2.0

Figure 18.3

0

35

70 Temperature (°C)

105

140

Compressive yield stress vs. temperature for Diab foams (Thomas et al., 2004 © Sage).

The polyurea/polyamide network in Diab HD 130 foam is responsible for the considerable compressive yield stress at 130°C (Fig. 18.3), well above the Tg of the PVC network. SEM of the cell structure of PVC foams of density 130 kg m⫺3 (Fig. 18.4) shows the cell face fraction φ is the order of 0.8.


(a)

(b)

Figure 18.4

(a) Microstructures of HD130 and (b) H130 Divinycell PVC foams (Kanny et al., 2004 © Sage).

5

PMI

7

PUR

6 5

PVC

4 3 2 1 0

Shear strength (MPa)

Compressive strength (MPa)

8 PMI

PVC

4 PUR

3 2

PIR

1

PIR 0

50

(a)

Figure 18.5

100

150

200

Density (kg m−3)

250

0

300

0 (b)

50

100

150

200

250

300

Density (kg m−3)

(a) Compressive yield stress vs. density graphs and (b) shear modulus vs. density graphs, for a number of foams (redrawn from Gundberg, 2002). PIR: poly isocyanurate.

Viana and Carlsson (2002) studied Diab H foams of densities between 60 and 300 kg m⫺3. They measured the tensile Young’s modulus and yield stress, then used the Gibson–Ashby models for closed-cell foams (Chapter 11) to deduce that the fraction φ of the polymer in the edges was between 0.5 and 0.6. However they did not examine the foam microstructure or realise that φ is likely to increase with the foam density. Figure 18.5a compares compressive yield stress vs. density graphs for a number of foams, while Figure 18.5b compares shear yield stress vs. density graphs. Polymethacrylamide (PMI) is a cross-linked, high Tg glassy polymer. Although PMI foams have the highest yield strength for a given density, their cost is also high. Such


431

foams, which resist high temperatures, are mainly for fibre-composite skins which are cured under pressure at temperatures above 70°C.

18.2.3 Skin materials Glass fibre reinforced thermosetting plastics (GRP) have a much higher design strain (about 1%) than aluminium (about 0.1%). Thin sheets of GRP can be bent to a small radius without failure occurring in either the fibres or the matrix. The sandwich core must resist any chemicals in the resin and have adequate mechanical properties at the curing temperature of the system. The applied pressures during the cure depend on the type of composite. For wet layup system, vacuum bag moulding applies a pressure of less than 100 kPa. However, composite prepreg layers are normally used with matched metal moulds and pressures the order of 1 MPa, to achieve adequate consolidation. The pressure exerted from the moulds is limited by the crush strength of the foam.

18.3 Elastic design of sandwich panels 18.3.1 Simple bending stiffness theory The critical design parameters of sandwich panels are usually their bending and torsional stiffness. The lightweight foam core increases the second moment of area of the skins, thereby increasing the panel bending stiffness, without a large increase in total mass. However the skins must be bonded to the core. If a metal skin is used with a thermoplastic foam, a separate adhesive layer is required. The foam surface is rough on the scale of the cell diameter, but aluminium skins must be pretreated for bonding strength. It is preferable if polymerisation of either the polyester resin in the GRP skin, or the PU in the foam core, bonds the sandwich together. The simple analysis assumes that the bonded skins and core bend to a common radius of curvature. The assumed Euler beam theory applies to long slender beams in which end effects can be neglected. Consequently, the longitudinal strain varies linearly with the distance from the neutral surface, and the longitudinal stress distribution is shown in Figure 18.6. Integration, across the beam cross-section, of the internal moments of these forces leads to an expression for the bending stiffness in terms of the Young’s moduli E and second moments of area I of the two components [ EI ]sandwich ⫽ ES IS ⫹ EC IC

(18.1)

432 Polymer Foams Handbook y Skin

Core

Neutral surface Stress σx

Skin

Figure 18.6

Cross-section of sandwich beam, with neutral axis and thin skins, and the longitudinal stress variation.

If the skins are relatively thin compared with the core, the panel bending stiffness is ⎛ h ⎞2 wh3 [ EI ]sandwich ≅ 2AS ⎜⎜ ⎟⎟⎟ ES ⫹ E ⎜⎝ 2 ⎟⎠ 12 C

(18.2)

The skin contribution tends to dominate the sum. The central deflection δ of a beam under 3-point bending, with distance L between the loading points, and central load 2F, is given by δ ⫽

3FL EI

(18.3)

Minimisation of the mass of a sandwich beam, of specified bending stiffness, indicates the use of a very low density, thick foam core. If particular materials are used for the skin and core, only the relative thicknesses can be varied. Murthy et al. (2006), among others, show that the bending stiffness to weight ratio is a maximum when the core mass is twice the skin mass. Increasing the skin separation increases the skin tensile (or compressive) strain at a given panel bending deflection, so makes better use of the skin tensile and compressive strength. In most applications, failure must also be avoided, and there may be limits on the panel thickness. Consequently, core optimisation is reconsidered later.

18.3.2 Bending stiffness of finite length panels A sandwich beam of finite length can bend both by extension/contraction of the skin length and by core shear. One extreme is a short


433

beam with a ‘soft’ core – Figure 18.7 shows a beam with half-length 4 times the core thickness (40 mm), under 3-point loading with a central deflection of 10 mm. The PS foam core has density 35 kg m⫺3, Young’s modulus 10 MPa, and Poisson’s ratio 0.1, while the 2 mm thick GRP skins have an in-plane Young’s modulus of 20 GPa and Poisson’s ratio 0.3. The core is assumed to stay elastic. The deformed skins have similar shapes, with very little extension, so the majority of the beam deflection is due to core shear. The core shear stress is close to 0.3 MPa over a large area; it reduces to zero both at the free end where no external shear forces are applied, and at the mid-beam, where it is zero by symmetry. The shear stress distribution differs from that in a single material beam (a parabolic variation with the distance from the neutral surface, being maximal there and zero at the upper/lower surfaces). The bending stiffness of this beam is less than 1% of that calculated using equations (18.2) and (18.3). Large deformation finite element analysis (FEA) was used for the simulation; this is essential as beam geometry changes affect the stress distribution. The tensile stiffness (force/extension) of a skin of thickness t and Young’s modulus Es, on a beam of half-length L and width W, is Eswt/L. If there were uniform shear stresses in the half-core, its shear stiffness would be GcwL/b, where Gc is its shear modulus and b its thickness. The ratio Γof the skin tensile stiffness to the core shear stiffness is given by Γ⫽

Es bt Gc L2

(18.4)

For the example in Figure 18.7a, Γ ⫽ 9.0. If Γ ⬎ 1, core shear dominates the deformation. For the beam in Figure 18.7b (with t ⫽ 0.2 mm, b ⫽ 10 mm, and Es ⫽ 40 MPa, but other parameters unchanged) Γ ⫽ 0.056. Although the deformation pattern approaches that assumed in the last section, the bending stiffness is still only 43% of that computed using equations (18.2) and (18.3).

18.3.3 Simple torsional stiffness theory We start by analysing the torsional stiffness of thin walled tubes of non-circular cross-section. The shear stress τ (Figure 18.8) acts parallel to the free surface of the tube. The ‘shear flow’ has a constant value τt around the perimeter of the tube, so the shear stress varies inversely with the wall thickness t. The total torque T acting on the tube is the sum of the moments about C of the shear forces F acting on the area elements dA of the section. Each produces a moment dT equal to Fh, so is equal to the shear


0.1

Free end

0.3

0.3 Symmetry at mid-beam (a)

0.15 0.1 0.0

0.2 (b)

Figure 18.7

Core shear stress contours (MPa) in sandwich beams at 10 mm deflection for Γ: (a) 9.0 and (b) 0.056 (Mills, unpublished).

F = τt ds t ds h

dA

C Core Included area A

(a)

Figure 18.8

(b)

Shear stresses in the cross-section of: (a) closed-hollow beam and (b) sandwich beam under torsion.

flow τt times twice the area dA of the shaded triangle. Hence the total torque T is given by T ⫽ 2τ t A

(18.5)

where A is the area inside the tube. Assuming the material is elastic, with shear modulus G, the tube twists by an angle θ when the torque is applied, and the stored elastic energy U ⫽ 1/2 Tθ. The stored


435

elastic energy U is also an integral of the energy density τ2/2G over the volume of the tube wall. For a length L of tube, using equation (18.3) to substitute for τ, it is U ⫽

L 兰 τ 2t ds ⫽ L 2G 2G

∫

T2 ds 4 A2 t

(18.6)

Rearranging equation (18.6), the torsional stiffness TL/θ is given by 4 A2 G 4 A2Gt TL ⫽ ⫽ ds θ p ∫ t

(if t is constant)

(18.7)

where p is the tube perimeter. Hence the torsional stiffness is proportional to the square of the included area A, to the skin thickness, and to its shear modulus. The skin of a sandwich panel does not run around the sides. Consequently, the core shear modulus affects shear load transfer from skin to skin. For sandwich panels that are wider than their thickness, if the core shear stiffness is adequate, a side region of the foam acts as a side skin (Fig. 18.9b). Foam cores, due to their low shear modulus G, contribute little to the panel torsional stiffness. The core shear stresses (except at the side regions) increase linearly with the distance y from the mid-beam. The torsional stiffness for a core of thickness t and width p (⬎⬎t) is TL ⫽ GJ ≅ 0.3Gpt 3 θ

(18.8)

where J is the core polar moment of area. Martinez and Sugura (2000) showed that the beam length affects its torsional stiffness. If the beam ends are built-in (encastre) the crosssection cannot warp near the ends, so the torsional stiffness increases.

18.3.4 Elastic deformation due to localised forces on the skin When localised loads are applied to a skin, the problem is similar to that of a beam on an elastic foundation analysed in civil engineering. Figure 18.9 shows the distribution of principal compressive stress in the sandwich beam analysed in Figure 18.7a. Although the principal stress directions are vertical near the beam mid-length, they are roughly at ⫾45° midway between the loading points, where core shear dominates. The stresses are highest near the centre loading points, but the 2 mm thick GRP skins are quite effective at spreading the load from the 6 mm radius cylinder.


0.8 0.6

0.1

0.4

0.2 0.4

0.2

Figure 18.9

Compressive principal stress contours (MPa) for the case shown in Figure 18.7a. (Mills, unpublished).

800

3500 Experimental data

Compressive strain in face

700

3000

Shear in core

2500 Shear in core

500

Load (N)

Load (N)

600

400

2000 Compressive strain in face

1500

300 1000

200

Compressive stress in core Compressive stress in core

500

100 Experimental data 0 0

2

4

6

8

10

12

Core thickness (mm)

(a)

14

0

16

0

(b)

2

4

6

8

10

12

14

16

Core thickness (mm)

Figure 18.10 Predicted and experimental failure loads vs. core thickness, for PU foam cores of densities (a) 96 and (b) 320 kg m⫺3 (Kim and Swanson, 2001 © Elsevier).

18.4

Failure modes The geometry of sandwich beams accentuates the shear and compressive loads on the core. Given the high tensile strength of skins and their ability to bend to a small radius of curvature, core failure modes may become critical. Kim and Swanson (2001) predicted that failure modes occur at loads that are functions of the core thickness. Figure 18.10 compares the predicted variations for a PU foam core of densities 96 and 320 kg m⫺3 with experimental data (the beam had CFRP skins of 0.5 mm thickness, the overall span of the 3-point bend test was 152 mm). Experimental data shows the failure mode change with face failure more likely with denser cores, and core shear more likely with lower density cores and thicker cores. These graphs would change if the loading geometry changes. Details of the failure modes are given the next three sections.


437

Figure 18.11 Shear failure in a sandwich beam with a 6 mm thick PU core of density 96 kg m⫺3 (Kim and Swanson, 2001 © Elsevier).

18.4.1 Core shear failure Localised core shear yield has little effect on the integrity of sandwich beams. However, if core shear yield spreads along the beam from the central loading point to the outer loading points, the yielding allows the skins to shear relative to each other and collapse is likely. After yield the core shear stress only increases slowly with increased strain, so further hardening is minor. Many of the core foams used in boat sandwich panels will fail, by forming cracks in echelon in directions perpendicular to the maximum principal stress, when the shear strain exceeds some value. This type of failure occurs in short beams loaded in 3-point bending, that is when the panel shear loads are high compared with the bending moments. However, for larger panels without free ends, it is less likely to occur. If the foam does not crack, there is little macroscopic change in the foam appearance post-failure. However, Figure 18.11 shows cracking in a core after a shear failure.

18.4.2 Core crushing due to surface loads Foam cores must resist compressive forces applied normal to the skin surface; if the skin sinks locally near the load application point, it is likely to buckle under the compressive stresses in the skin. Thin skins have a limited load spreading capacity, so the foam compressive collapse stress is important. The geometry of the load application affects the results; although it is easier to consider load application via cylinders or spheres, sandwich panels will be subjected to a variety of loading geometries. Figure 18.12 repeats the simulation of Figure 18.7a, for an EPS foam core with a compressive yield stress of 0.29 MPa. The contours of plastic strain magnitude near the cylindrical load application point indicate core crushing in this region. This yielding allows the foam core to crush in thickness, consequently there is elastic local bending of the upper skin. This stopped the FEA simulation at an 8.5 mm beam deflection.

438 Polymer Foams Handbook 0.25 0.15 0.05

Figure 18.12 Contours of plastic strain magnitude in the core, for the case of Figure 18.7a, at 8.5 mm beam deflection (Mills, unpublished).

Magnitude of residual dent (mm)

4 Theory Test ABAQUS

3.5 3 2.5 2 1.5 1 0.5 0

0

2

4

6

8


Figure 18.13 Magnitude of residual dent vs. indentation magnitude for H60 PVC foam core (Zenkert et al., 2004 © Elsevier).

Zenkert et al. (2004) considered localised crushing of Rohacell WF51 and Divinycell H60 foam cores when protected by a GRP skins, when the load was applied by a cylindrical steel roller of 25 mm diameter. The residual dent was plotted against the indentation magnitude (Figure 18.13). FEA overestimates the size of the residual dent, but FEA could not simulate core damage under the load point. Better predictions were achieved with a multistage analytical theory, in which part of the face sheet was detached over the crushed core and then reattached after the core was allowed to relax. Rizov et al. (2005) considered a 25 mm diameter spherical indentor pressing on the surface of a sandwich panel, the lower surface of which was supported by a rigid flat table. The core was 50 mm thick PMI foam of density 52 kg m⫺3 and the fibreglass skins had thickness 2.4 mm. At loads exceeding 1.5 kN, the foam crushed and skin damage was noted. FEA considered foam yield, but assumed the GRP skin remained elastic. It could predict the diameter of the damaged area, but it slightly overestimated the force deflection relationship (Fig. 18.14).


439

Damage diameter (mm) 0

20

40

60

80

100

120

5

Load (kN)

4

3

FEA

Test

2

1

0

0

1

2

3 4 Deflection (mm)

5

6

Figure 18.14 Load indentation curves (test data and FEA) plus damage zone diameter (䊊 data, dashed curve FEA) (redrawn from Rizov et al., 2005 © Elsevier).

It appears therefore that better modelling of GRP damage is required to improve the modelling of the complete sandwich beam. Core crushing can lead to compressive failure of a skin, usually by buckling. If the skin delaminates from the core, a planar crack is formed under the skin. Aviles and Carlsson (2006) compute the effect of such defects on the in-plane buckling loads of sandwich panels with GRP skins and PVC foam cores. There is typically a 50% reduction in the buckling load if a circular debond of diameter 70 mm forms below one skin.

18.4.3 Core cracking Shear failure of the core foam may lead to cracks forming. Alternatively, when the sandwich skin is impacted, it can often bend and rebound without damage. If the foam core is crushed, when the skin rebounds, the core foam is placed under tension and penny-shaped cracks can initiate in the foam close to, and parallel to, the skin. When core cracks grow, they cannot propagate through the skins, so most turn to be parallel to the skins. Consequently, large core cracks cause delamination of the skins from the core and weaken the panels. Consequently, the fracture toughness of core foams is important (Chapter 15).

440 Polymer Foams Handbook 18.4.4 Core creep The foam will creep under any sustained load, and this will probably be the major factor in the creep of the sandwich panel. When panels are used beneath the concrete floors of buildings, there is a compressive creep load on the foam. Specifications for the static stress are given in BSEN 13164 (2001). Surprisingly, the design loads for concrete floors are rather low. The thick floor slab protects the foam from localised damage. Taylor et al. (1997) measured the bending creep of SIP panels of EPS and PU cores. They were concerned with the shape of the creep curve, and gave no creep compliance data. The creep strain for the EPS cored SIP seems to almost linearly increase with time. Since the skins are wooden veneer, there can be contributions to the overall creep from the skins under tension and compression, as well as from the foam core under shear.

18.5

Sport applications Skis are sandwich beams, with complex, multilayer skins. Details of ski design are shown in Federolf (2005). For bending stiffness, it is important to control bending vibrations. Surfboards and windsurfer boards use foam cores with an external skin of fibre-reinforced thermoset or a thermoplastic moulding. If the core is to be used as the ‘mandrel’ for curing a thermoset, the foam must resist the solvents such as styrene present. Thus EPS is not suitable for such constructions.

18.6

Thermal insulation panels 18.6.1 Insulation panel design Modern building regulations require thermal insulation, both under concrete floors and in walls and roofing, to reduce the energy consumption of homes and industrial buildings. The panels used in walls or roofs are only lightly stressed – there are bending stresses from the self-weight, plus wind or snow loads. If they are beneath concrete floor slabs, then the design stress relates to the floor loading plus the mass of the concrete cover. For thermal insulation panels, the skin role in reducing foam cell gas loss by diffusion may be more important than its mechanical roles. Metallic film layers can be bonded to the surfaces of the panels to reduce the diffusion of the cell gas. However this is not totally effective. The gas


Table 18.2

441

Thermal conductivity of common blowing agents

Gas O2 N2 CO2 Isobutane Isopentane HFC-245a HCFC-141b CFC-11

Molecular weight (g mol⫺1)

Thermal conductivity at 10°C (mW m⫺1 K⫺1)

32 28 44 58 72 102 117 137

24.9 24.6 15.3 14.8 13.7 12.3 9.1 7.4

transfer coefficient at the surface of the panel decreases by roughly a factor of 10 compared with the uncoated panels, but not to zero.

18.6.2 Thermal requirements Glicksman (1994) considered three main contributions to the thermal conductivity of closed-cell foams: the thermal conductivities of the polymeric structure and the gas, and the radiation contribution. In most insulating foams, the gas contribution is the largest. In general, gas thermal conductivity decreases with increases in molecular weight (Table 18.2, taken from Randall and Lee (2002)). Therefore gases with a high molecular weight but high boiling point are required. When chloro-fluorocarbon (CFC) blowing agents were banned, foams were manufactured using other gases with low thermal conductivities. A vacuum can be used for special applications (Manini, 2001) but this places a large mechanical load on the open-cell foam structure. Typical 1.5 µm thick faces in PU foams are nearly transparent in part of the infrared (IR) spectrum, so some radiation can pass long distances through the foam structure. However the edges in PU foams absorb and re-radiate IR radiation; there is a complex theory for the radiation scattering. An integrating sphere system is normally used to determine the IR contribution to foam conductivity. Figure 18.15 compares measurements with five theories for the radiation effects; none of these makes exact predictions, but the data for PU foams increases almost linearly with the Z average of the cell diameter. The thermal conductivity of the polymeric structure depends on its geometry and dimensions. Models should consider the edge and


Radiative conductivity (mW m−1 k−1)

10 B 8 D

6

C 4 E A

2

0

0

200 400 Cell dimensions (µm)

600

Figure 18.15 Radiation contribution to thermal conductivity vs. cell diameter: data for PU foams (X closed cell, ◊ open cell) vs. theories: B Cunningham and Sparrow, C Glicksman edges and faces, D Glicksman edges only (Eeckhaut and Cunningham, 1996 © Sage).

vertex geometry (Chapter 1). Ahern et al. (2005) noted that there was a significant vertex contribution to the electrical conductivity of foams, which is governed by the same equations. However the thermal conductivity of air is about 10% of that of solid PU, whereas the electrical conductivity of air is negligible, so the thermal conductivity problem is more difficult. Hence the theory is semi-empirical, using shape factors for the edges and faces, and including a vertex contribution. Thermal conductivities predicted by their model are compared with the thermal conductivities of PU foams (minus the radiation term) in Figure 18.16; this shows the benefit of minimising the foam density. The microstructure of EPS, with the majority of the polymer in the cell faces, differs from that of PU foam. This increases the IR scattering from cell faces and reduces the radiation contribution to the overall conductivity. Figure 18.17 shows that the thermal conductivity of EPS foam is a minimum at densities about 40 kg m⫺3; the increase at lower densities is due to the increasing radiation contribution. The higher minimum value for EPS compared with PU foams reflects the higher thermal conductivity of air compared with hydro-chlorofluorocarbons, butane or pentane. The measured thermal conductivity of EPS only reaches a constant value when the foam block is greater than about 100 mm thick, because

Gas + solid thermal conductivity (mW/m K)


443

34 Experimental (Total − Radiation) Maxwell + vertex

33 32 31 30 29 28 27

3

3.5

4

4.5 5 5.5 6 Solid volume fraction (vol%)

6.5

7

Figure 18.16 Gas plus solid thermal conductivity vs. foam relative density data for PU foams: compared with theory (Ahern et al., 2005 © Elsevier).

0.036

λe (W m−1 K−1)

0.035 0.034 0.033 d /µm = 300

0.032 0.031

d /µm = 150

0.030 0

10

20

30

40

50

ρ (kg m−3)

Figure 18.17 The variation of the thermal conductivity of EPS foam with density and cell diameter (Quenard et al., 1998 © Pion).

there is a greater radiation contribution in the surface layers than deep inside the block (BASF, 2001).

18.6.3 Aging of PU foam thermal insulation with skins The thermal conductivity of PU foams increases with time due to the high-conductivity air diffusing into the cells then the slower diffusion


Thermal conductivity (Btu in./h ft2 °F)

0.20

0.18

0.16 90°F 40°F −10°F

0.14

0.12 0

10

20

30

Time1/2 (days)1/2

Figure 18.18 Predicted (curve) and measured aging of 10 mm slices of PU foam insulation (Wilkes et al., 2002).

of blowing agents from the cells. The phenomenon is related to the gas diffusion contribution to creep (Chapter 19) and water diffusion (Chapter 20). Gas diffusion from large panels occurs in the direction normal to the panel surface, since the lateral extent is much larger than the typical 50–100 mm thickness. Hence the results can be rationalised using the Fourier number, a dimensionless time Fo ⬅

Dt L2

(18.9)

where D is the diffusion constant for the gas in the foam and t is the time. For a slab of foam that loses gas from both sides, L is the half-thickness. Results for any slab thickness and any diffusion constant can be compared using concentration profiles at given values of the Fourier number. If a 10 mm slice from an insulating panel is tested, a particular diffusion profile develops in 1/25 of the time that it would take for a 50 mm thick panel. Fan (1997) studied the aging of PU panels and considered some of the two- and three-dimensional diffusion problems at their edges. Wilkes et al. (2002) studied the aging of PU foam thermal insulation used in refrigerators. The foam is typically 50 mm thick, with an inner liner of ABS or high impact polystyrene (HIPS), and an outer steel skin. In order to accelerate the PU foam aging, slices of foam 10 mm thick were aged at three temperatures. The subsequent thermal conductivity of the slice was plotted against the square root of time (Fig. 18.18). For a single gas diffusion process, this would be expected to provide a linear graph, but there is diffusion of CO2 and O2/N2


445

from the air at different rates. The model predictions for these experiments were not perfect, but better agreement could be obtained by adjusting the diffusion constant for oxygen. When simulation refrigerator panels were tested, the nature of the 1 mm thick thermoplastic skins had an effect. The permeation constant for CO2 at 4°C, was 9 units [cc(STP)/100 in.2atm day] for the ABS but 56 units for the HIPS. The corresponding figures for O2 were 1.9 and 8.5 units for ABS and HIPS respectively. Therefore, not surprisingly, panels with two ABS skins showed a 7% increase in thermal conductivity after 3 years at 4°C, while those with HIPS skins had a 14% increase. The authors did not give the rubber volume fraction in these polymers, which would be expected to affect the permeability.

Summary Foams cores are preferable to honeycomb cores for building panels, due to the ease of bonding to complex skin shapes, and the low thermal conductivity of foam. However, in some structural applications, metal honeycombs are preferable due to their higher shear moduli and compressive strength parallel to the cell axis. Given the difficulty of analysing the material behaviour of fibrecomposite skins and foam cores for the relatively simple geometry of a sandwich panel, it is no wonder that the analysis of motorcycle helmets, with a curved fibre-composite outer shell, not bonded to a EPS liner, is also complex (Chapter 16).

References Ahern A., Verbist G. et al. (2005) The conductivity of foams: a generalisation of the electrical to the thermal case, Coll. Surf. A 263, 275–279. Aviles F. & Carlsson L.A. (2006) 3-D FEA of buckling of debonded sandwich panels, J. Composit. Mater. 40, 993–1008. BASF (2001) Styropor Technical Information, Ludwigshafen, Germany. BS EN 13164 (2001) Thermal insulation products for buildings – factory made products of extruded polystyrene (XPS) specification. Danielsson M. (1996) Toughened rigid foam core material for use in sandwich constructions, Cell. Polym. 15, 417–435. Eeckhaut G. & Cunningham A. (1996) The elimination of radiative heat transfer in fine celled PU rigid foams, J. Cell. Plast. 32, 528–552.

446 Polymer Foams Handbook Fan Y. (1997) The effect of heat and mass transfer on the cellular plastics insulation and long-term aging, Thesis, VTT, Finland. Federolf P.A. (2005) Thesis, at e-collection.ethbib.ethz.ch/ecol-pool/ diss/fulltext/eth16065.pdf. Glicksman L.R. (1994) Heat transfer in foams, Chapter 5, in Low Density Cellular Plastics, Eds. Hilyard N.C. & Cunningham A., Chapman and Hall, London. Gundberg T. (2002) Foam cores in the marine industry, on boatdesign.net. Hartsock J.A. (1991) Design of Foam Filled Structures, 2nd Edn., Technomic, Lancaster, PA. Jackson J.E., Weaver M. & Highsmith A.L. (1999) Materials considerations for high speed ships, Proceedings of 5th International Conference on Fast Ships, Seattle, Washington, DC. Kanny K., Mahfuz H. et al. (2004) Static and dynamic characterization of polymer foams under shear loads, J. Composit. Mater. 38, 629–639. Kim J. & Swanson S.R. (2001) Design of sandwich structures for concentrated loading, Composit Struct. 52, 365–373. Koschade R. (2002) Sandwich Panel Construction, Ernst & Sohn, Berlin. Manini P. (2001) Recent developments in open-cell polyurethane vacuum filled insulation panels for super insulation applications, in Advances in Urethane Science and Technology, Eds. Klempner D. & Frisch K.C., RAPRA, Shawbury. Martinez F. & Sugura J.M. (2000) Effects of non-uniform torsion in homogeneous or composite beams, Conference: Mathematical Theory of Networks and Systems, Perpignan, France. Murthy O., Munirudrappa N. et al. (2006) Strength and stiffness optimization studies on honeycomb core sandwich panels, J. Reinf. Plast. Composit. 25, 663–671. Quenard D., Giraud D. et al. (1998) Heat transfer in the packing of cellular pellets: microstructure and apparent thermal conductivity, 14th ECTP Proceedings, 1089–1095. Randall D. & Lee S., Eds. (2002) The Polyurethanes Book, J Wiley, New York. Rizov V., Shipsha A. & Zenkert D. (2005) Indentation study of core sandwich composite panels, Composit. Struct. 69, 95–102. Taylor S.B., Manbeck H.B. et al. (1997) Modelling structural insulation panel flexural creep deflection, J. Struct. Eng. ASCE 123, 1658–1665. Thomas T., Mahfuz H. et al. (2004) High strain rate response of crosslinked and linear PVC cores, J. Reinf. Plast. Composit. 23, 739–749. Viana G.M. & Carlsson L.A. (2002) Mechanical properties and fracture characterization of cross-linked PVC foams, J. Sandw. Struct. Mater. 4, 99–113.


447

Wilkes K.E., Yarborough D.W. et al. (2002) Aging of polyurethane foam insulation in simulated refrigerator panels – 3-year results with third generation blowing agents, J. Cell. Plast. 38, 317–338. Zenkert D. (1997) The Handbook of Sandwich Panel Construction, EMAS, Caradley Health, UK. Zenkert D., Shipska A. & Persson K. (2004) Static indentation and unloading response of sandwich beams, Composites Part B 35, 511–522.

Chapter 19

Modelling of creep and viscoelasticity

Chapter contents 19.1 Introduction 19.2 Creep in solid polymers 19.3 Creep in polymer foams 19.4 Micromechanics of open-cell foam creep 19.5 Cyclic loading of open-cell foams 19.6 Creep in closed-cell foams 19.7 Discussion References

450 452 454 460 466 467 476 477


19.1

Introduction 19.1.1 Viscoelasticity in foam applications Sections 5.3.6–5.3.8 introduced the main viscoelastic phenomenon. Creep loading was met in connection with flexible seating foams (Chapter 9) and rigid foam packaging for electronic goods (Chapter 12). If a polyurethane (PU) foam car seat creeps, the driver’s eye position sinks with respect to the rear-view mirror; and the mirror needs to be adjusted. In orthopaedic seating and mattress applications, there is creep loading for periods of hours. Both heating and moisture absorption from the body may accelerate the creep rate. Foam hardening at high compressive strains is important for the design of seating, since high strains occur in some regions of cushions. Hysteresis in cyclic loading affects how automotive foam seats damp out vibrations from rough road surfaces. Stress relaxation is of minor importance. ‘Geofoam’ layers of large extent, up to 1 m thick, are used in the construction of road embankments across weak soil (bogs, etc.), replacing denser rocks or soil in embankments. The low foam density means that the pressure of the embankment on the sub-strata is not excessive. However, the high volume of foam used means that its cost is critical. Certain properties of low-density foams are similar to those of weak soils. For instance, the uniaxial compressive yield stress of both can be the order of 20 kPa. However, under a hydrostatic compressive stress, the foam changes volume, whereas the soil has a near-constant volume. For a soil density ρ 艑 2500 kg m3, acceleration of gravity g 9.8 m s2, the thickness of the top layer h 艑 0.5 m, and the creep stress on the foam ρgh 艑 12 kPa. Some allowance is then made for dynamic vehicle loads. In a conservative design, the foam strain is kept below 1%. If the foam compressive strain exceeds 10%, the loss of air from the cells may add to the creep mechanisms.

19.1.2 Viscoelastic models Viscoelastic theories are mathematical models (Ferry, 1970) for material behaviour. The parameters used are not linked to the polymer microstructure or to the foam structure. In creep loading, a constant stress σ, usually applied at time zero, causes a time-dependent strain ε(t) where the time t 9. Compressive loading is most common for foam. The creep compliance J is defined by J(σ, t) ≡

ε(t) σ

(19.1)

Chapter 19 Modelling of creep and viscoelasticity

451

In general J is a function both of stress and time, and the material response is described as non-linearly viscoelastic. When the strains are small, the creep compliance function becomes independent of stress σ; the creep strain, for a linear viscoelastic solid, is given by ε(t) σJ(t)

(19.2)

to distinguish between solid polymers and foams, the compliances are written JP(t) and JF(t), respectively. It is often easier to measure the creep compliance function J(t) than the relaxation modulus E(t), because it is easier to apply a constant load than a constant strain in a fraction of a second. Linear viscoelasticity theory can relate E(t) to J(t). Unless the polymer is near a transition temperature, the creep compliance is a slowly varying function of time, so 1 E(t)

J(t) ≅

(19.3)

Stress relaxation data can be modelled by a set of Maxwell viscoelastic elements (a spring representing a Young’s modulus Ei, in series with a dashpot representing a viscosity ηi) connected in parallel, plus a single elastic element E for the long-time response (Fig. 19.1). The model has a relaxation modulus n

E(t) E∞ ∑ Ei exp( t /τ i )

(19.4)

i 1

E1

η1

Figure 19.1

E2

η2

E3

η3

Ei

E∞

ηi

Maxwell elements in parallel with a single spring used to model stress relaxation.

452 Polymer Foams Handbook where τi are relaxation times, defined as ηi /Ei. The mathematical form of equation (19.4) is a Prony series. The constants can be obtained by curve-fitting stress relaxation data. The model can then be used to predict the material response to creep, a steady strain rate, sinusoidally varying strain, or any other strain history.

19.2

Creep in solid polymers 19.2.1 Polyurethanes For foam creep modelling, viscoelastic data is required for the polymer in the microstructural state that it has in the foam. It may be difficult to obtain, since the orientation of thermoplastic cell faces, or the ‘crystallinity’ of a PU, often differs from that of solid specimens. Most crosslinked PU foams cannot be remoulded into a solid slab. Until data is available from micro-measurements on foam elements, it is necessary to use data from related solid PU. Zhu and Mills (1999) made stress relaxation measurements on a solid PU cylinder, moulded from a lightly crosslinked PU foam of high density. It was compressed to a fixed strain in 8 s, then the stress monitored for the succeeding 5000 s, during which time the stress falls by about 25%. The relaxation modulus E(t), when plotted against log t, is nearly a straight line (Fig. 19.2). The data can be approximated by equation (19.4), using relaxation times of 0.1, 1, 10, . . . , 105 s with moduli E1 5, E2 E3 6, E4 E5 7, E6 E7 9.3 MPa, plus

Relaxation modulus E(t) (MPa)

80 70 60 50 40 30 20

1

10

100

1000

104

105

Relaxation time (s)

Figure 19.2

Stress relaxation data for solid PU, moulded from Bulpren foam, for an initial strain of 0.89% (Zhu and Mills, 1999).


453

E 20 MPa. At time zero, the sum of the moduli is E(0) 102.4 MPa. When stress relaxation was carried out for a range of initial strains up to 11%, and the 100 s relaxation modulus E(100) plotted against the strain, the non-linear behaviour was described by E(100) 64.05 3.68ε 0.125ε2

(19.5)

where ε is the strain and the modulus is expressed in MPa.

19.2.2 Polyethylene, ethylene–vinyl acetate, and polypropylene The microstructure of the oriented semi-crystalline polymer in foam cell faces is not spherulitic. However, for low-density polyethylene (LDPE) foams, the polymer viscoelastic response appears to be similar to that in bulk mouldings. Consequently, Mills and Gilchrist (1997a) determined the creep compliance of solid crosslinked polyolefins at 25°C, prior to the dissolution of the nitrogen gas in the Zotefoams process. The fraction crystallinity and final melting temperatures were determined by differential scanning calorimeter (DSC) (Table 19.1). The crosslinked ethylene–vinyl acetate (EVA) copolymer had a vinyl acetate content of 18%. Creep compliance curves for solid LDPE superimposed for stresses up to 1.4 MPa, showing that the polymer was in the linear viscoelastic region. They can be described by JP JP(1)(1 B log10 t)

(19.6)

where Jp(1) is the 1 s polymer creep compliance and B is a constant (Table 19.2). An alternative power-law equation JP JP(1)tn

(19.7)

fitted data for a solid EVA better than equation (19.6). The creep of a solid polypropylene (PP) copolymer (Table 19.2) supplied by Zotefoams plc was measured in the linear viscoelastic region by Mills (1997). Its creep compliance curve is typical for a PP.

Table 19.1

Polymers tested by Mills and Gilchrist (1997) and Mills (1997) Polymer LDPE EVA PP

Density (kg m3) Crystallinity (volume %) 917 940 889

32 19 43

Melting temperature (°C) 104 81 153


19.3

Creep in polymer foams 19.3.1 Open-cell flexible PU foams Compressive creep measurements on open-cell PU foams (Campbell, 1979; Phillips and Auslander, 1992) revealed near-linear graphs of strain vs. the logarithm of time. Campbell attributed the maximum in the creep rate (dε/d log t) at a strain of 20–30% to cell-edge buckling, a process which apparently began at 10% strain. The creep rate reduced significantly at strains exceeding 60%. These results were confirmed by Moreland et al. (1994). Zhu and Mills (1999) measured the compressive creep of Bulpren S PU foam (Fig. 19.3). For a low stress of 1 kPa the creep strain remains a linear function of log time. For a creep stress of 2 kPa, the slope increases when the strain reaches 10%, although the maximum slope does not occur until 104 s. For creep stress of 3 kPa and above, the creep rate starts to decrease at 50% strain. For low creep stresses, the foam creep compliance is independent of the creep stress, that is the behaviour is linearly viscoelastic. There are three regimes of creep compliance behaviour: 1. For strains less than 5%, the foam creep compliance JF(t) is proportional to the polymer creep compliance JP(t). 80 9 70 60

5

Strain (%)

50 4 40 3

30

2.5

20

3

2

10 1 0

Figure 19.3

0

1

2 3 Log time (s)

4

1 5

Creep at 25°C for Bulpren foam, compressed across the rise direction, with the stresses in kPa: solid lines data and dashed lines predictions for R 0.025 for the stress relaxation data of equation (19.4) (Zhu and Mills, 1999).


455

2. For strains between 5% and 60%, there is a higher creep rate, as a result of reorientation and bending of the edges. 3. For strains exceeding 60%, contact between edges reduces the bending moments and the creep rate reduces.

19.3.2 Slow-recovery PU foam Davies and Mills (1999) investigated Confor 47 slow-recovery foam of density 93 kg m3 with a microstructure similar to Fig. 1.6. The maximum creep rates at 25°C in the second creep regime (Fig. 19.4) were higher (80% compressive strain per time decade) than for conventional flexible PU foam (17% per decade), hence there was a quicker transition, from a collapse strain of 10% to an edge-contact strain of 60%. The Confor creep curves at 45°C were similar in many ways to the 25°C flexible PU foam creep curves, suggesting similar creep mechanisms.

19.3.3 LDPE and EVA closed-cell foams Mills and Gilchrist (1997a) tested 70 kg m3 density LDPE foam (LD70) at 25°C. The family of creep curves (Fig. 19.5) diverged at times greater than 10 s; they will eventually converge at a compressive strain 艑80% at a time greater than 109 s.

100 90

60 kPa

80 15 kPa

Strain (%)

70

10 kPa

60 50 40 30 20

5 kPa

10 0 1

1 kPa

0

1

2

Log time (s)

Figure 19.4

Creep at 25°C for Confor 47 foam (Davies and Mills, 1999).

3


50

100

Strain (%)

40

79

30 20

110

80

10 0

68 60 48

130

1

2

3

4

5

6

Log time (s)

Figure 19.5

Compressive creep of LD70 foam at 25°C; stresses in kPa. Dashed lines simulations with σY(1) 170 kPa, B 0.33, εY 0.05, p0 117 kPa, ν 0, and air diffusivity 250 1012 m2 s1 (Mills and Gilchrist, 1997).

70 Compressive strain (%)

60 50

120 100

40 30

79 20 10 0

0

1

2

3

4

5

6

Log time (s)

Figure 19.6

Recovery after 6 105 s creep at 25°C at stresses in kPa indicated of LDPE foam of density 70 kg m3 (Mills and Gilchrist, 1997).

Recovery after creep is shown in Figure 19.6 for the three highest stress samples of Figure 19.5. If the recovery curves are inverted, they appear to be scaled versions of the 80 kPa creep graph, that is the time scale of recovery seems to be independent of the previous creep stress. Recovery after low stress creep is much faster; the yield strain is not reached so there is no gas loss from the foam. The creep compliance curves superimpose for times less than 1 s, but diverge at longer times if the strain greater than 5%, due to the


457

Table 19.2 Parameters for creep compliance curves, between times tmin and tmax, using equations (19.4) and (19.5).

Material

Stress (kPa)

Log10 tmin (s)

Log10 tmax (s)

J0 (109 m2 N1)

Equation (19.4) r

Equation (19.5) r

B

n

Solid LDPE Solid EVA LD70 foam LD24 foam EV35 foam

910 221 47.5 5.0 6.6

0.0 0.5 0.1 0.0 0.0

6.0 5.7 6.0 6.1 6.0

3.73 23.7 301 1523 2640

0.997 0.974 0.980 0.995 0.979

0.990 0.986 0.994 0.996 0.993

0.334 0.163 0.625 0.803 0.264

0.0759 0.0475 0.0968 0.1055 0.0627

The bold data indicates the equation with the highest correlation coefficient r, that fits the data best.

non-linear stress–strain relationship and gas escape from the foam. Table 19.2 shows that the foam compliance functions are best fitted by the power-law equation (19.5), while that of the solid polymer is best fitted by equation (19.4). There are systematic deviations from both equations at very short and long times. The slopes (B or n) of the compliance functions are higher for the foams than for the corresponding solid polymers. The slope of the foam creep graph increases with increasing stress; for EV35 foam at stress 3.3 kPa, n 0.0450, close to that for the solid polymer, but n 0.0627 at stress 6.6 kPa.

19.3.4 Polystyrene bead foam Horvath (1998) fitted data creep for expanded polystyrene (EPS) Geofoam using the Findley equation ε ε0 mtn

(19.8)

where ε0 is the instantaneous strain, n and m are constants, and t is time. However, it is difficult to measure ε0; it takes a finite time to apply the creep load, and, immediately afterwards, the creep strain increases rapidly. Horvath used an iterative process to find the value of ε0 that leads to a linear plot of ε ε0 vs. t on doubly logarithmic ‘axes. The initial Young’s modulus, calculated using ε0, was nearly independent of the creep stress at 5.0 0.5 MPa. Figure 19.7 compares data for EPS of density 20 kg m3 and extruded polystyrene (XPS) foam of a comparable 22 kg m3 density. It shows the marked effect of processing on the response. The XPS is anisotropic, with cells elongated in the direction of the applied creep stress, and lacks the inter-bead channels of the EPS, while the EPS is nearly isotropic. Consequently, the Young’s modulus of the XPS in the stress direction is higher and its creep strain is smaller at a similar stress.


EPS 100


12 10

159

8 140 6 4 81

EPS 50 2 0

0

1

2

3

4

5

6

Log10 time (s)

Figure 19.7

Comparison of compressive creep of EPS and XPS of density approximately 21 kg m3 at the stress levels (kPa) indicated (Mills, unpublished).

The creep data for the XPS is not in the linear viscoelastic region, since the creep compliance increases with the creep stress. When the EPS foam yields at a time circa 100 s for the 100 kPa stress, the creep rate accelerates faster than it does for the XPS. Gas diffusion to the bead boundaries may play a part in the collapse process. EPS of density 23 kg m3 will yield at a stress of 70 kPa after a time exceeding 1000 hours, as shown by the isochronous stress–strain graphs of Horvath (1994). EPS can be moulded in very large blocks and this, rather than the slightly inferior creep properties at a given density, is of major importance in Geofoam applications.

19.3.5 Polypropylene bead foam In expanded PP (EPP) foam the continuous air channels at the bead boundaries (Chapter 4) allow much faster gas escape than in extruded foam, because the maximum gas diffusion distance is the bead radius. For 43 kg m3 density EPP foam (Mills, 1997), the S-shaped creep curves diverge for times in excess of 1 s (Fig. 19.8). The steepest part of each curve occurs when the strain is about 40%; increasing the creep stress appears to shift a curve, of near-constant shape, horizontally to


459

100

Strain (%)

80

60

300 250 200 175 150

40

130 100

20

85 70

0

60 0

1

2

3

4

5

6

Log time (s)

Figure 19.8

Creep curves for 43 kg m3 density PP foam, labelled with creep stresses in kPa (Mills, 1997).

Table 19.3

Parameters for EPP creep compliance, between 1 and 106 s, using equation (19.5) Foam density (kg m3) 31 43 60 902 (solid)

Stress (kPa)

J0 (109 m2 N1)

n

r

9.9 20 60 1430

1600 411 307 0.697

0.033 0.0582 0.0464 0.0505

0.973 0.996 0.992 0.989

shorter times. The creep compliance curves only superimpose for low stresses of 20 and 40 kPa, when the strain remains less than 6%, that is before yield occurs. Equation (19.5) provides a good approximation to the creep compliance data for the 20 kPa stress level. The correlation coefficient r and the best-fit parameters are given in Table 19.3. The data for the 20 kg m3 density foam has a positive curvature, so no fitting parameters are given. If the creep strain exceeds 5%, the bending of cell edges and faces provides geometrical non-linearity, the polymer viscoelastic response may become non-linear, and air escape contributes to the creep strain. The foam compliance then does not superimpose for different creep stresses. When the creep load is removed, less than 10% of the maximum creep strain is recovered instantaneously. Figure 19.9 of the residual


175 150

Relative strain (%)

80

60

40

20

0

0

1

2

3

4

5

6

Log time (s)

Figure 19.9

Relative recovery of EPP foam of density 43 kg m3 after 6 105 s creep at 25°C at stresses in kPa indicated (Mills, 1997).

strain/maximum creep strain vs. time shows there is very little relative recovery for the 1st hour. The majority of the cell gas has escaped in the high strain creep, so this stage of recovery is driven by the viscoelasticity of the polymer, and opposed by the below-atmospheric cell pressure. When air diffuses back into the cells the recovery rate increases.

19.4

Micromechanics of open-cell foam creep 19.4.1 Low strain creep For foam creep strains less than 2%, the geometry change is insignificant, so it cannot cause non-linearity in the foam response. If the polymer is linearly viscoelastic, the foam behaves as a linear viscoelastic solid. An open-cell Kelvin foam of relative density R, made from a polymer of Young’s modulus EP, is predicted to have a Young’s modulus (Chapter 7) in the [111] direction given by EF CEPR2

(19.9)

where the constant C is 1.0 for compression in the [111] direction and 2.2 for compression in the [001] direction. The equivalent equation


461

for the creep compliance of a linearly viscoelastic foam, justified in the next section, is JF (t)

JP (t) CR2

(19.10)

Combining equations (19.2) and (19.10), the small strain creep behaviour of the Kelvin model can be described by ε(t)

σ0 CR2

JP (t)

(19.11)

Nolte and Findley (1970) invoked a ‘space frame’ model, consisting of a cubical array of struts, to explain the relationship between the open-cell foam and PU compliances. Huang and Gibson (1991) used a similar model, proposing cell-edge bending as the main creep mechanism. Assuming a linearly viscoelastic polymer, they used dimensional analysis to develop a scaling relationship between the foam creep compliance and R2. This is less useful than equation (19.10) since the scaling constant is unknown. Although they reported shear creep data for rigid PU foams at strains less than 5%, they did not provide any data for the solid PU to verify the relationship.

19.4.2 Analytical model of high strain creep During high strain creep, the bending moment distributions along cell edges change as the foam geometry changes, so both the stress and the strain in the polymer change with time. The effects of this geometric non-linearity are considered; the initial stages of an analytical theory are expounded, then finite element analysis (FEA) models, which confirm the analysis, are explained. The analytical theory works for any applied stress, whereas, so far, FEA simulations were limited to moderate creep stresses. Zhu and Mills (1999) approximated the foam cell edges in a Kelvin foam as Euler–Bernoulli beams (Chapter 7) with constant crosssections. The foam was compressed in the [001] direction, so only deformation mechanism was edge bending. The model does not consider interactions between touching edges, so it cannot predict the reduction in the creep rate at strains greater than 60%. In the structural cell (Fig. 7.4) all the slanting cell edges have the same orientation with respect to the applied stress axis, and there is symmetry about the edge midpoint. Consequently, only the half-edge DO needs to be analysed. A constant vertical force F, applied to both ends of the edge, produces a time-dependent moment M(t) on the edge given by

462 Polymer Foams Handbook s

M(t) F ∫ sin θ(t)ds

(19.12)

0

where θ(t) is the time-dependent edge orientation angle. This external moment is equated to the moments of the longitudinal stresses σ acting on the edge cross-section s

M(t)

∫

A

σ(t)y dA F ∫ sin θ(t)ds

(19.13)

0

where dA is an element of the section area at a distance y from the neutral surface. Boltzmann’s superposition principle for linear viscoelastic materials relates the stresses to the material relaxation modulus E(t). Assuming that the polymer has been stress free for times t 0, the principle states that the stress is an integral of the strain rate history t

σ(t)

∫

0

dε E(t τ )dτ dτ

(19.14)

where τ is a dummy time variable for integration. Substituting this equation into (19.13) gives t

∫∫

A 0

s

dε E(t τ )dτ y dA P ∫ sin θ(t)ds dτ

(19.15)

0

The strain ε in the edge is a function of y and the curvature ε(y, t) y

dθ(t) ds

(19.16)

This is differentiated with respect to τ to give dε d ⎛⎜ dθ ⎞⎟ y ⎜ ⎟ dτ dτ ⎜⎝ ds ⎟⎟⎠

(19.17)

This can be substituted into equation (19.15), and the resulting integral separated, as the edge cross-section does not change on bending

Chapter 19 Modelling of creep and viscoelasticity t

463

s

∫

E(t τ )

0

d ⎛⎜ dθ ⎞⎟ ⎜ ⎟ dτ y 2 dA P ∫ sin θ(t)ds dτ ⎜⎝ ds ⎟⎟⎠ ∫ A

(19.18)

0

The second integral on the left side of equation (19.18) is the second moment of area I of the edge cross-section. Hence, the concept of I is still valid. When the equation is differentiated with respect to s, it yields an expression for the shear force on the edge t

I ∫ E(t τ ) 0

d ⎛⎜ d2θ ⎞⎟⎟ ⎜ ⎟ dτ F sin θ(t) dτ ⎜⎜⎝ ds2 ⎟⎠

(19.19)

Further algebra leads to an equation for the time-dependent response t

I ∫ E(t τ ) 0

⎛ ⎞ dθ ′′ E(t) dτ F ⎜⎜ sin θ(t) sin θ(0)⎟⎟⎟ ⎜⎝ ⎟⎠ dτ E(0)

(19.20)

where θ is shorthand for d2θ/ds2. θ is approximated by a finite difference scheme, and the half-edge divided into 100 segments. A recurrence relation gives the change in θ in a time interval, in terms of the relaxation of the generalised Maxwell model. The predicted edge shape when the foam strain is 40%, after 2 105 s of creep, is similar to one calculated for elastic deformation (Fig. 7.9). The PU foam tested had a relative density R 0.025. For a creep stress of 1 kPa, the predicted and experimental creep strains are similar; both are predicted to increase nearly linearly with log time (Fig. 19.3). The solid PU creep compliance function J(t) is nearly linearly proportional to log time, so the foam and solid PU creep curves have the same shape, as predicted by equation (19.11). The 3 kPa experimental curve at t 1 s has a similar strain to the prediction, but the experimental creep rate accelerates between 10 and 100 s, suggesting that there is more geometric non-linearity in the PU foam than in the model; hence the foam and solid PU creep curves differ in shape. The constant C in equation (19.10) is deduced as 1.0, as expected for the uniform edge cross-section model.

19.4.3 FEA of high strain creep Mills (unpublished) considered the creep response of a wet Kelvin foam, compressed in the [001] direction, using a linear viscoelastic model in standard FEA with 3D elements. The boundary conditions were a constant force applied to a larger representative unit cell (RUC) (Fig. 7.11) than that considered in the last section. The simulations were

464 Polymer Foams Handbook 0.012 4 3 2 0.8

Creep compliance (kPa−1)

0.011

0.01

0.009

0.008

0.007

0.006

1

10

100

1000

104

Time (s)

Figure 19.10 Predicted creep compliance curves for wet Kelvin foam compressed in the [001] direction at the labelled stresses (kPa) (Mills, unpublished).

only successful for creep stresses up to 5 kPa, in a model with R 0.027, due to computational instability at higher stresses. The predicted compliance curves (Fig. 19.10), for the set of shear moduli and relaxation times given in Section 19.5, almost superimpose, showing there is relatively little geometric non-linearity for strains of up to 5.5%. Although the shape of the compliance curves is the same as that predicted by Zhu and Mills (1999), the compliance values are lower because the constant C in equation (19.10) is 2.2 for the wet Kelvin model. These simulations confirm the analytical approach described in the last section. They suggest that a model with a greater geometric non-linearity is required to explain the experimental data for PU foams. A model for the creep of a wet Kelvin foam compressed in the [111] direction could provide such geometric non-linearity. However, this has not yet been implemented, because of the difficulty of applying a constant total force to a number of nodes subject to periodic boundary conditions. Therefore, the over-simple Gibson–Ashby open-cell model with R 0.026 (Fig. 7.5) was analysed to see whether the three main regions of foam creep response (Section 19.3.1) could be predicted. When this model was compressed at a constant strain rate, viscoelastic FEA predicted a near plateau in the compressive stress– strain curve as certain edges buckle. In the creep model, the RUC was compressed between large rigid anvils, preventing deformed edges from entering the neighbouring RUC, hence simulating edge-to-edge contact. For a creep stress of 1 kPa, the creep strain remains below 0.2 as


465

0.6

Compressive strain

0.5

0.4 6 0.3

4 2

0.2

1 0.1

0

1

10

100

1000 104 Creep time (s)

105

106

Figure 19.11 Predicted creep of the Gibson–Ashby model compressed in the [001] direction, at the stresses (kPa) indicated (Mills, unpublished).

the ‘vertical’ edges never buckle (Fig. 19.11). This graph is similar to that for Confor foam under a 5 kPa stress in Figure 19.4. For higher creep stresses, the graphs progressively increase in slope until certain edges contact the rigid anvils at a strain of 0.5. The maximum creep rates are considerably higher than those measured experimentally (Fig. 19.3). However, small changes in cell geometry would affect the collapse stress. This indicates the importance of cell size and shape variations in ‘spreading’ the creep response over a wide time scale. After the initial edge-to-edge contact, creep continues at a reduced rate, since the foam is not fully densified. Edge contact in irregular foams occurs over a much wider strain range, and the first edges to undergo large deformation can penetrate undeformed neighbouring cells. Slow-recovery foams show marked creep at 20°C in the time range 0.1–100 s due to the closeness of Tg. The appropriate relaxation time spectrum, with high values of gi over part of the time spectrum, could be used to model such behaviour.

19.4.4 Non-linear viscoelasticity and cell anisotropy For compressed PU foams of relative density circa 0.025, the analysis in Section 7.6 shows the maximum polymer strains are of the same

466 Polymer Foams Handbook order as the macroscopic foam strain. Therefore, the polymer viscoelastic response may be non-linear. However, foam micromechanics modelling has not considered non-linear viscoelasticity. Gibson and Ashby (1997) considered a power-law relationship between stress and strain rate, in a model in which cell edges are loaded axially in compression. They concluded that there is a critical stress below which edge buckling will not occur, although they did not produce any experimental evidence that this mechanism change occurs in polymer foams. Most PU foams have anisotropic cell shapes, as a result of a sheet of foam rising before the polymerisation is complete. The elastic response of an anisotropic version of the Kelvin foam was analysed by Gong et al. (2005). Such a model could also be used for creep analysis. Although the regular Kelvin foam is not ideal for modelling foam creep, it has been the most successful so far in predicting creep responses.

19.5

Cyclic loading of open-cell foams

6

6

5

5

4

4

Stress (kPa)

Stress (kPa)

FEA of [111] direction compression of a wet Kelvin foam model for an elastic material (Section 7.6.2) predicted that both edge bending and twisting occur. Mills (2006b) used a viscoelastic material in implicit FEA for this model, to simulate cyclic compression testing. PU material properties were a combination of elasticity, with short-time Young’s modulus E0 50 MPa and Poisson’s ratio ν0 0.45, and viscoelasticity, with a Prony series of dimensionless shear moduli gi at relaxation times 0.01, 0.03, 0.1, 0.3, 1, 3, and 10 s. Figure 19.12 shows the predicted compressive stress–strain, for a strain cycle to

3 2

(a)

2nd

3 2 1

1 0

1st

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Compressive strain

0 (b)

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Compressive strain

Figure 19.12 Cyclic compression stress–strain curves: (a) for PU foam of density 26 kg m3 and

(b) predicted for Kelvin foam in the [111] direction. Solid graph for all gi 0.1 and dashed graph for all gi 0.05 (Mills, 2006b).


467

74% compressive strain in 1 s. The loading stress–strain graph for the viscoelastic model has more negative curvature than the elastic model, and the unloading response falls below the loading response. The hysteresis, defined as the energy lost in a loading/unloading cycle, divided by the energy input, increases with the maximum strain in the cyclic tests. When the model was reloaded, the second loading curve falls below the initial one, a phenomenon noted with PU foams. The predicted unloading curve for all gi 0.1 is closer to the experimental data than that for all gi 0.05, in spite of the latter being a better representation of the solid PU viscoelasticity. In neither case is the predicted hysteresis (Table 19.4) as high as the experimental values.

19.6

Creep in closed-cell foams The modelling of creep in closed-cell foams is less advanced, compared with that for open-cell foams. If the creep stress is insufficient to cause yield, the creep strain is low and there is no geometric nonlinearity. Consequently, the linear viscoelastic response of the foam is proportional to that of the polymer, and the contribution from the compressed cell gas is low. Until the modelling of yield in the polymer structure (Chapter 11) is improved and the deformation mechanisms fully established, it is premature to attempt a complete micromechanics analysis of closedcell foam creep. Nevertheless, the strain rate dependence of the compressive yield stress of some polymer foams has been characterised. Clutton and Rice (1991), analysing isochronous stress–strain graphs, deduced that the gas pressure in foamed PE and PP declined during creep, as air diffused through the cell faces and escaped. Therefore, by modelling the effects of a time-dependent yield stress, and gas losses by diffusion, progress was made in modelling closed-cell foam creep (Mills and Gilchrist, 1997a). It was assumed that the polymer contribution was independent of the foam strain, and that gas and polymer act in parallel. The contributions are considered in the following two sections.

Table 19.4

Hysteresis of foam cube, compared with wet Kelvin model predictions, for maximum compressive strain 75% Foam and density

Hysteresis (%)

PU 27 (kg m3) Model gi 0.05 Model gi 0.10

60 17 40

468 Polymer Foams Handbook 19.6.1 Time-dependent foam yield stress The yield stress of LDPE foams in Instron tests was defined as the intersection of the initial loading curve and a straight line drawn for the post-yield part of the curve, giving a yield strain of 10%. For creep tests, the yield stress is the creep stress, and the yield time was that when the creep strain reached 10%. The two sets of (yield stress and yield time) data for LDPE foam are not consistent (Fig. 19.13), partly because the creep test temperature was 5°C higher than that for the Instron tests, and partly because the strain rate varied in the creep tests but was constant in the Instron tests. Assuming that the LDPE foam is linearly viscoelastic, with a compliance given by equation (19.5) with J(1) replacing JP(1), the yield strain εY is reached after a time tY, where σY

εY J(1)(1 B log t Y )

(19.21)

The creep yield data (Fig. 19.13) is fitted using B 0.33 from the solid LDPE, and the 1 s creep yield stress σY(1) 170 kPa. Figure 19.13b shows that the initial yield stress of PP foam decreases almost linearly with the logarithm of time, for times greater than 1 s. For room temperature tensile tests on isotactic PP film (Alberola et al., 1995), the yield stress varied linearly with the logarithm of strain rate, but there was a steep upturn when the strain rate exceeded 200 s1. In Figure 19.13b the initial yield stress also rises for a yield times less than 0.1 s. Both phenomena are due to the time dependence of the glass transition of the amorphous phase (10°C at low strain

Instron 150

100

50 (a)

400 Initial yield stress (kPa)

Yield stress (kPa)

200

Creep yield

0

1

2 3 4 5 Log time to yield (s)

300 200

(b)

43 31

100 20 0 −2

6

60

−1

0

1 2 3 Log time (s)

4

5

6

Figure 19.13 Compressive yield stress of foam at 25°C vs. log time to yield. (a) LD70: solid line equation (19.21) with σY(1) 170 kPa and B 0.33 (Mills and Gilchrist, 1997). (b) EPP creep yield data with foam densities in kg m3 (Mills, 1997).


469

rates). This is further evidence that the foam compressive yield involves the (tensile) yield of the cell walls.

19.6.2 Gas diffusion from the compressed foam Analytical solutions exist for gas diffusion in undeformed foam (Section 18.6.3). However gas diffusion in highly deformed foam is a coupled problem, since the stress affects the pressure in the cells and the creep strain affects the diffusion coefficient. The cell faces act as barriers to gas flow, while the diffusion of gas across the cells is rapid. The cells act as constant pressure gas reservoirs, with pressure gradients across the cell faces, while the gas seeks the easiest route through the network of cell faces. Equation (11.14) relates the foam diffusivity Df to the foam relative density R and the polymer permeability P via Df

6Ppa φR

(19.22)

where φ is the fraction of polymer in the cell faces and pa is the atmospheric pressure. The nitrogen permeability P of Dow Chemical LDPE film at 24°C (Anon, 1995) is 4.5–9.1 1013 m3 at STP m1 s1 atm1. LDPE foam of density 22 kg m3 has a relative density R 0.024. Table 19.5 gives LDPE foam diffusivities, using the measured φ 0.9. For EVA copolymer film having an 18% vinyl acetate content, the oxygen permeability is 435 cm3 at STP mm m2 day1 atm1, and the nitrogen permeability is about 1/3 of this. Extrapolating from the diffusion constant of 46 pm2 s1 for oxygen in LDPE (Mills, 2005), the value for EVA foam of density 275 kg m3 is approximately 100 p m2 s1. The diffusivity of foamed glassy polymers is much lower. EPS foam of density 35 kg m3 has a nitrogen diffusivity of 6.5 1012 m2 s1 (Schwartz and Bomberg, 1989). The experimental values in Table 19.6 are estimates from modelling the foam creep

Table 19.5

Foam diffusivities Df at 25°C

Polymer LDPE LDPE EVA EVA

Density (kg m3)

Predicted Df for N2 (1012 m2 s1)

66 22 34 275

42–83 125–250 260 100

Experimental Df for air (1012 m2 s1) 250 500 1000 100

470 Polymer Foams Handbook curves. Verdejo (2003) measured a value of 70 1012 m2 s1 for EVA foam of density 152 kg m3. This and the measured 480 1012 m2 s1 value for PE foam of density 20 kg m3 (Section 3.3.3) show that the estimated values in Table 19.6 are of the correct order of magnitude.

19.6.3 Modelling compressive creep Air diffuses from foam cells towards the closest free surface, hence for cube-shaped samples, the majority of the air diffusion is in a direction perpendicular to the stress axis. Mills and Gilchrist (1997) used a finite difference method to solve this problem. However, in many products (shoe soles, foam insulation below concrete foundations) which involve foam sheet compressed perpendicular to the sheet surface, the gas diffusion direction is parallel to the stress axis; this case will be explained here. Mills and Rodriguez-Perez (2001) found that cells close to the lower surface of EVA foam flatten after creep, suggesting that cell gas diffuses in the direction of stress (the y-axis in Fig. 19.14a). In this series load transfer model, the applied stress is constant throughout the foam,

Centre

Stress

line Cell 1 Gas diffusion

Cell 2 L

y

Cell n−2 Cell n−1

Stress

Polymer contribution

Loaded Cell n

Gas contribution Strain

(a)

Prior to loading

0.7

1− R

(b)

Figure 19.14 (a) Schematic of gas diffusion, parallel to the stress axis, through a compressed foam with cubic cells. (b) Assumed contributions of the gas and polymer to the total stress (Mills and Gilchrist, 1997).


471

while the strain varies from cell to cell in the y direction. The applied pressure is shared between the polymer structure and the pressurised cell gas. Under creep loading, when the compressive strain is between 0.1 (above yield) and 0.7 (before faces touch), there is assumed to be a constant polymeric contribution (Fig. 19.14b) at a particular time. Hence there is a constant gas pressure p in the cells, independent of their y co-ordinate. The pressure in any closed cell, in contact with the loading plates, must remain at p until most of the cell gas has escaped. The modelling considers interactions between the cell strain, gas diffusivity, cell face touching at high strains, and the polymer strain rate. Initially all cells have a gas content pcell(n) 100 kPa, the cell pressure if the cell is returned to zero strain. This is also the percentage of the initial cell gas content. A sequence of six steps is repeated at a time interval dt small enough for stable calculations: (a) Diffusion, using a finite difference recurrence relationship for the pressure pnew in cell n, of initial diameter L, after an interval dt pnew(n) (1 2D) pold(n) D(pold(n 1) pold(n 1)) dt

L2 2Df

and

D

max

(19.23)

Df (n) 2Df

(19.24)

max

where Dfmax is the maximum of the cell diffusivities Df(n). (b) The cell gas content pcell(n), from the previous value multiplied by pnew(n)/pold(n). (c) The absolute pressure pabs applied to the cell gas in the foam is related to the creep time t by pabs σ pa σY(t)

(19.25)

where σ is the creep stress and σY(t) is given by equation (19.21). (d) The strain ε(n) in each cell, using ⎛ p (n) ⎞⎟ ε(n) (1 R) ⎜⎜⎜1 cell ⎟⎟ ⎜⎝ pabs ⎟⎠

(19.26)

(e) The progressive contact of cell faces, once the foam strain exceeds a limit (0.7), causes the percentage of stress carried by the polymer to increase, reaching 100% when the foam strain is 1 R. The absolute pressure of each cell was pold(n) Gpabs. The facetouching factor G decreases linearly, from 1 at 0.7 strain to 0 at strain 1 R (Fig. 19.14b).

472 Polymer Foams Handbook (f) The diffusivity of the compressed cell Df(n), from the cell strain

Df (n)

Df

0

(19.27)

1 ε(n) R

The foam diffusivity was not allowed to increase by a factor of more than 10. Undeformed EVA foam of density 275 kg m3 was assumed (Table 19.6) to have D 100 pm2 s1. The foam Poisson’s ratio 0.0 and the cell diameter 0.1 or 0.2 mm, while the foam block was 16 mm thick. A high strain develops in the foam cells close to the loading anvils for times greater than 10,000 s (dashed curves in Fig. 19.15), due to positive feedback between diffusivity and strain increases. The majority of the foam is at about 40% strain, but four cells, closest to the surface, are at about 70% strain, close to the limiting value of 1 R. The solid curves show that the cell gas content is 100 kPa in the central part of the foam, but 15 kPa or less in the surface cells. There is a sudden gas content change across a couple of cells then a small gradient across the depleted cells. When the number N of cells with less than 50 kPa gas pressure is plotted against time t on logarithmic scales, the best-fit equation is N 0.0457t0.525

(19.28)

100

100 3 80 6

5

4

60

60 3

40

40

20

20 6

0

Strain (%)

Cell gas content (kPa)

80

0

1

2 3 Distance from midplane (mm)

5 4

4 5

0

Figure 19.15 Predicted cell gas content and layer strains vs. y co-ordinate for the series model, for log10 (creep time s) indicated. Cell diameter 0.1 mm and creep stress 200 kPa (Mills and Rodriguez-Perez, 2001).


473

This suggests a depletion rate proportional to the square root of time, a result expected from Fickian diffusion processes. In the predicted creep graphs (Fig. 19.16), the creep rate slows when the creep strain reaches 0.74, but the graph separation for creep stresses from 100 to 200 kPa is greater than for experimental data. The predictions do not change if the cell size is increased from 0.1 to 0.2 mm. The higher the creep stress the shorter the time for the foam strain to reach 70%. The effect of cell face touching (step (e) in the model) is critical to the results. If it is ignored, the gas content of the outer cells falls to zero, preventing the escape of further gas. Initial experimental measurements with a low-density EVA foam (Verdejo, unpublished) confirm the development of densified foam layers, close to the loading surfaces.

19.6.4 Data for closed-cell PE foams Mills and Gilchrist (1997) established that the sample size and foam density affected the rate of gas loss from LDPE foams under creep loading. However the effects were not examined systematically; large samples were cut into slices, so the sample shape changed with its thickness. Rodriguez-Perez et al. (2006) made systematic studies of LDPE foams, evaluating the diffusivity by analysing isochronous stress–strain graphs in the way described in the next section. When the sample size was kept constant at 40 mm by 40 mm by 30 mm, the diffusion 75 500 400

60

Strain (%)

350 45

250

250

200

200 30 150

150 15

100 100

0 1

2

3 4 Log time (s)

5

6

Figure 19.16 Creep of EVA foam of density 275 kg m3 at the marked stress levels (kPa): solid lines experimental data and dashed lines model (Mills and Rodriguez-Perez, 2001).

474 Polymer Foams Handbook coefficient of crosslinked LDPE foams was a linear function of the reciprocal foam density (Fig. 19.17), a result predicted by equation (19.22). They found that the diffusion coefficient was a decreasing function of the side of a cube sample (Fig. 19.18).

Diffusivity (10−10 m2 s−1)

4

3

2

1

0 0.01

0.02

0.03

0.04

0.05

Reciprocal density (m3 kg−1)

Figure 19.17 Diffusivity of crosslinked LDPE foam at 23°C as a function of the reciprocal density. A linear relationship is predicted by equation (19.22) (Rodriguez-Perez et al., 2006 © RAPRA).

Diffusivity (10−10 m2 s−1)

3

2.5

2

1.5

1

0.5 10

20

30 40 Cube edge (mm)

50

60

Figure 19.18 Diffusivity of crosslinked LDPE foam at 23°C as a function of sample edge length (Rodriguez-Perez et al., 2006 © RAPRA).


475

19.6.5 Data for closed-cell EPP foams Figure 19.19 shows the creep stress plotted against the gas volumetric strain for 20 kg m3 density foam, for a constant creep time of 105 s. This isochronous data fits a straight line, if the stress exceeds the initial yield stress. Isochronous plots were made for a range of creep times, and for impact data, for which a time of 0.01 s was used (the stress level changes with time, so the impact data is not really compatible with the creep data). The fitting parameter p0, the effective gas cell pressure, is shown in Figure 19.20 as a function of the creep time. The two low-density foams have nearly the same gas pressure variation with time. The gas loss rate is slower by a factor of 50 for extruded LDPE foam of density 24 kg m3, than it is for 20 kg m3 density PP foam, in spite of LDPE having 3 times the permeability to oxygen of PP. However, air diffuses from EPP beads of diameter 4 mm, rather than from a LDPE cube of 25 mm side. The larger diffusion distance should delay the gas loss from the LDPE foam by a factor of (25/4)2 40. A possible complicating factor is that gas only escapes after the foam yields, and the time dependence of the LDPE foam yield stress is slightly different from that of the EPP copolymer foam (Fig. 19.13). As the density of EPP foam increases from 20 to 60 kg m3, the foam diffusivity should decrease by a factor of 3, if the bead size and the foam yield stress were both constant. The bead diameter only decreases slightly as the foam density increases (Table 4.2). The gas escape from the two high-density foams appears to be delayed by a factor of 10 in time, compared with the low-density foams (Fig. 19.20).

500

Creep stress (kPa)

400

300

200

100

0

0

2

4

6

8

10

12

14

16

Gas volumetric strain after 105 (s)

Figure 19.19 Isochronous stress vs. gas volumetric strain for PP foam of density 20 kg m3 at a creep time of 105 s, fitted with equation (11.4) (Mills, 1997).

476 Polymer Foams Handbook 250 60 Gas pressure p0 (kPa)

200

43

150 20 100 31 50

0

0

1

2

3 Log time (s)

4

5

6

Figure 19.20 Gas pressure vs. creep time for EPP foams of density indicated (Mills, 1997).

19.7

Discussion Polymer viscoelasticity dominates the low strain (5%) compressive creep response of closed-cell polyolefin foams, whereas cell gas loss dominates the high strain creep. The analysis (Chapter 7) of the high strain compression of wet Kelvin foams with relative densities between 0.02 and 0.06 shows the maximum tensile strains in parts of cell edges is 5%, when the foam is compressed by 5%. Hence the polymer will be in the non-linear viscoelastic region, or will yield, and the creep mechanism will change at about 5% foam strain. Semi-crystalline polyolefins have much higher gas diffusivities than glassy polymers such as polystyrene (PS), so gas loss contributes more to high strain creep in the former foams. However, if the foam initial yield stress exceeds 200 kPa, the gas escape contribution becomes less important. Hence high-density polyethylene (HDPE) or PP foams may be preferred to LDPE foam, if other mechanical properties such as toughness stay the same. Foam creep test specifications (ASTM, 1984) do not acknowledge the effect of sample size on the results; the preferred sample size is 6 in. by 6in. by 4 in., with a minimum size of 2 in. by 2in. by 1 in. As packaging often uses much smaller blocks of foam, in which there can be faster gas escape, there is a risk of excessive creep in such foam if ASTM data is used for design. Low-density foams used for buoyancy aids (Chapter 20) could perform badly if these products are stored under a significant creep load, and the gas is allowed to escape. Foam recovery on unloading is rather slow; as the only driving force is


477

unbending of the cell edges and faces, the foam density will remain high for some hours. This contrasts with their good recovery after compressive impact, when the high cell gas pressure causes the strain to return quickly to zero.

References Alberola N., Fugier M. et al. (1995) Tensile mechanical behaviour of quenched and annealed isotactic polypropylene over a wide range of strain rates, J. Mater. Sci. 30, 860–868. Anon (1995) Permeability and Other Film Properties of Plastics and Elastomers, Plastics Design Library, New York. ASTM D 2221-84, Creep Properties of Package Cushioning Materials, American Society for Testing and Materials, PA. Campbell G. (1979) Compressive creep of flexible polyurethane foams, J. Appl. Polym. Sci. 24, 709–723. Clutton E.Q. & Rice G.C. (1991) Structure–property relationships in thermoplastic foams, Cellular Polymers Conference, RAPRA Technology Ltd, Shawbury. Davies O.L. & Mills N.J. (1999) The rate dependence of Confor PU foams, Cell. Polym. 18, 117–136. Ferry J.D. (1970) Viscoelastic Properties of Polymers, Wiley, New York. Gibson L.J. & Ashby M.F. (1997) Cellular Solids: Structure and Properties, 2nd edn., Cambridge University Press, Oxford, pp. 242–245. Gong L., Kyriakides S. & Jang W.Y. (2005) Compressive response of open-cell foams, Part 1. Morphology and elastic properties, Int. J. Solid. Struct. 42, 1355–1378. Horvath J.S. (1994) Expanded polystyrene geofoam: an introduction to material behaviour, Geotext. Geomembr. 13, 263–280. Horvath J.S. (1998) Mathematical modeling of the stress–strain– time behaviour of geosynthetics using the Findley equation. Research report available at www.engineering.manhattan.edu/civil/faculty/ Horvath/ Huang J.S. & Gibson L.J. (1991) Creep of polymer foams, J. Mater. Sci. 26, 637–647. Mills N.J. (2006) Finite element models for the viscoelasticity of open cell polyurethane foam, Cell. Polym. 25, 277–300. Mills N.J. (2005) Plastics: Microstructure and engineering applications, 3rd Ed., Butterworth-Heinemann, Oxford, p. 326. Mills N.J. (1997) Time dependence of the compressive response of polypropylene bead foam, Cell. Polym. 16, 194–215. Mills N.J. & Gilchrist A. (1997) Creep and recovery of polyolefin foams-deformation mechanisms, J. Cell. Plast. 33, 264–292.

478 Polymer Foams Handbook Mills N.J. & Rodriguez-Perez M.A. (2001) Modelling the gas-loss creep mechanism in EVA foam from running shoes, Cell. Polym. 20, 79–100. Moreland J.C., Wilkes G.L. & Turner R.B. (1994) Viscoelastic behaviour of flexible slabstock polyurethane foams: dependence on temperature and relative humidity, Part I. Tensile and compressive stress relaxation, Part II. Compressive creep behaviour, J. Appl. Polym. Sci. 52, 549–568, 569–576. Nolte K.G. & Findley W.N. (1970) Relationship between the creep of solid and foam polyurethane resulting from combined stress, Trans. ASME J. Basic Eng. 92, 106–114. Phillips J.C. & Auslander J. (1992) Compressional behaviour of inked open-cell foams, Polym. Eng. Sci. 32, 668–677. Rodriguez-Perez M.A., Ruiz-Herrero J.L. et al. (2006) Gas diffusion in polyolefin foams during creep tests, Cell. Polym. 25, 221–236. Verdejo R. (2003) Gas loss and durability of EVA foams used in running shoes, Ph.D. thesis, University of Birmingham, Birmingham. Zhu H.X. & Mills N.J. (1999) Modelling the creep of open-cell polymer foams, J. Mech. Phys. Solid. 47, 1437–1457.

Chapter 20

The effects of water

Chapter contents 20.1 Introduction 20.2 Phenomena 20.3 Theory of water uptake kinetics 20.4 Open-cell foams and sponges 20.5 Closed-cell bead foams 20.6 Closed-cell foams References

480 480 482 485 490 496 500


20.1

Introduction Foams come into contact with water in a number of applications. Hydrophilic foams, used as sponges, contrast with foams used for buoyancy or for thermal insulation, require near-zero water absorption. Bead foams are intermediate cases, where the slow water uptake can affect thermal insulation or reduce the mechanical strength. The phenomena are discussed first, prior to the relevant diffusion theory. Finally, water-related issues for the three main foam types are discussed.

20.2

Phenomena 20.2.1 Types of water absorption in open-cell polymer foams The water in uncompressed foam can be categorised as either: (a) Excess, draining from the foam when it is lifted from water. (b) Capillary, held in the cells by capillary action; some can be ejected by compressing the foam. (c) Hygroscopic, existing as molecular layers on the polymer surface. (d) Absorbed, held by hydrophilic forces inside the swollen polymer. For typical polyurethane (PU) or PVA (polyvinyl alcohol) foams, contribution (c) can be ignored. The other contributions are analysed in the following sections. Synthetic sponges are used as models of porous soils, when teaching water retention to geographers and civil engineers (Saskatchewan University, 2003). However, while compressing a sponge can expel some water, it is not easy to compress the volume of soils.

20.2.2 Draining under gravity The hydrostatic pressure at a distance h below the surface of water is ρgh, where ρ is the density of water and g the acceleration of gravity. The equilibrium height hE of water in a vertical capillary tube of radius r is hE ⫽

2γ cos θ ρ gr

(20.1)

where γ is the surface tension of the water/gas interface and θ the wetting angle. Gomes et al. (1997) used contact angle measurement to determine the surface tensions of polyester/polyether PU foams of densities 28–45 kg m⫺3 as being in the range 25–30 mJ m⫺2.

Chapter 20 The effects of water

481

It is not clear how to adapt equation (20.1) for open-cell foams. Chen et al. (1999) used it to determine the effective capillary radius of a PU foam, and found this to be the same order of magnitude as the cell diameter seen by scanning electron microscope (SEM). Cooper (1996) doubled the constant in equation (20.1) without giving a justification. Water will drain from foams removed from water, if the hydrostatic pressure at the foam base exceeds the capillary attraction, hence its height exceeds hE. The empirical Darcy law, for the steady flow at volumetric rate Q (m3 s⫺1) of a liquid of viscosity η, through a porous solid of permeability k, under a pressure gradient ∆P/L Q⫽

kA⌬P ηL

(20.2)

was met in Chapter 8. This equation, assuming streamline flow, can be used to analyse the rate of water drainage from foam.

20.2.3 Effect of pressure on the water content of flexible open-cell foams Cooper (1996) assumed that the foam modulus was unaffected by the presence of water. His incomplete analysis apparently assumes that water is expelled by mechanical strain, with no contribution from surface tension forces. He estimated the pressure applied by hand to a wiper to be the order of 1 kPa, while the hydrostatic pressure at the lower surface of 60 mm thick foam would be 0.6 kPa. This hand pressure will only expel significant water if the foam modulus is less than 10 kPa; however, he gave no modulus data for commercial wiper foam.

20.2.4 Water absorbed by open-cell PU foam Articles in this area, written by physicists and chemical engineers, describe phenomena and analyse rates of water uptake. In some articles, details of the PU foam chemistry and mechanical properties are lacking, so it is difficult to link the phenomena to the foam microstructure. PU foams can be hydrophilic or hydrophobic, and their microstructures range from open-cell through part-open to closed-cell, so the possible water uptake rates vary enormously. Sabbahi and Vergnaud (1993) studied the weight gain when a sheet of PU foam of density 510 kg m⫺3 was immersed in boiling water; at this density the microstructure would consist of isolated spherical bubbles. The diffusion constant was 6.5 ⫻ 10⫺11 m2 s⫺1 for the first absorption but 32 ⫻ 10⫺11 m2 s⫺1 for two subsequent absorptions. In

482 Polymer Foams Handbook a related article (1991) they found the diffusion constant for the foam to be anisotropic.

20.3

Theory of water uptake kinetics 20.3.1 Driving mechanisms The driving forces for water uptake should be identified before seeking to analyse the phenomenon. Diffusion theory can be applied to the closed-cell foams, where diffusion is the only method of water entry. For bead foams, the primary route of water ingress is flow in the inter-bead channels (Section 20.5.2), followed by diffusion into the beads. For open-cell foams, there is water permeation into the pore space, driven by a combination of hydrostatic pressure and surface tension forces at the polymer/water interface, followed by diffusion into the polymer. The theory for the permeability of connected cells in open-cell foams, and bead boundary channel networks in expanded polystyrene (EPS), is related to those for pore networks in rocks. The porosity of open-cell foams is close to 1, while it is ⬍0.05 for most bead foams. In some polymers, such as PVA, water absorption reduces the glass transition temperature Tg to below room temperature, changing the polymer from a glassy state to a rubbery state, modifying the absorption kinetics.

20.3.2 Kinetics of mass change of a foam slab When water diffuses into foam, diffusion theory can predict the concentration gradient. This analysis can either be applied on the microscale to foam edges/faces or on the macro-scale to blocks of foam. Some of the geometries of interest are shown in Figures 20.1 and 20.2. On the macro-scale, foams are treated as continuum solids with effective diffusion constants. Such analyses should be considered before planning experiments on water uptake. The shape of the predicted mass increase vs. time graphs assists the interpretation of experimental data. If the predictions are inadequate, a model with more than one diffusion coefficient may be necessary. The theory for diffusion, from a slab or cube of material with a constant diffusion coefficient α, occurs in many textbooks, for example Luikov (1968). For a slab, diffusion occurs in one direction. The dimensionless mass increase θ is given as a function of the immersion time t by

␪⫽

M(t) ⫺ M0 ⫽ Mf ⫺ M0

⬁

∑ Bn exp(⫺␮n2 Fo)

(20.3)

n ⫽1

where M(t) is the time-dependent mass, M0 the original mass, and Mf the mass after an infinite immersion time. The constants Bn and µn in


0.9 0.8

483

0.9 0.8 0.7 0.6

0.7

0.5

0.6

0.4

0.5

0.3

0.4

0.3

Cylinder axis

Mirror plane

(a)

Figure 20.1

Mirror plane

Mid-height

(b)

Iso-concentration contours, at Fo ⫽ 0.3, on the internal symmetry surfaces of (a) cube and (b) cylinder with height ⫽ 2 ⫻ diameter (Mills, unpublished).

1

0.9 0.7 0.5 0.3

Normalised water content

0.9 0.8

Figure 20.2

Cube Plateau border

0.7 0.6 0.5 0.4 0.3

(a)

Cylinder

0.1

1 Fourier number

(b)

(a) Iso-concentration contours of a Plateau border at Fo ⫽ 0.3. (b) Normalised water uptake vs. Fourier number for three geometries (Mills, unpublished).

the Fourier series are given by Bn ⫽

2 8 ⫽ 2 2 µn π (2n ⫺ 1)2

(20.4)

484 Polymer Foams Handbook The Fourier number Fo is defined by Fo ⬅

ατ L2

(20.5)

where α is the diffusion coefficient and τ the immersion time. For a slab of foam in contact with water on both sides, L is equal to the half thickness of the slab.

20.3.3 Mass change of a foam cylinder or cube Although the approach of the previous section gives results for diffusion into a cube, it is also possible to use finite element analysis (FEA) to compute mass diffusion. The water diffusivity and solubility in the foam must be specified. Figure 20.1 shows, for a cube of side L, and a cylinder of height 2L and diameter L, iso-concentration contours at a Fourier number of 0.3. The Fourier numbers are computed using equation (20.5), while the concentration is made dimensionless by division by the solubility of water in the foam. It is possible, by summing the variable ESOL (the concentration multiplied by the element volume) in ABAQUS, to calculate the total water absorbed as a function of the Fourier number. Figure 20.2b shows the variation of the dimensionless mass increase, defined in equation (20.3), for the two geometries; the responses are similar but not identical. With the Fourier number on a logarithmic scale, the graphs are nearly linear for Fo ⬍ 1, but they saturate for Fo ⬎ 4.

20.3.4 Mass change of a Plateau border Analytical solutions are not available for diffusion into a Plateau border edge, so FEA was used. The Fourier number was computed using L ⫽ b, the Plateau border width. The concentration contours at Fo ⫽ 0.3 (Fig. 20.2a) show that the projections of the Plateau border, with a high surface area to volume ratio, absorb water more rapidly than elsewhere. The dimensionless mass increase (Fig. 20.2b) is slightly less than for the other geometries at the same Fourier number.

20.3.5 Permeation theories for open-cell foams and bead foams Chapter 8 considered theories for the (air-flow) permeability of various foam structures. The predicted values can also be used for the water permeability of these structures. Equation (20.1) can be used with a measured polymer surface tension and foam mean cell diameter to


485

calculate the capillary pressure. However, there seem to have been no predictions of the rate of water ingress into open-cell or bead foams, based on the capillary pressure. There is likely to be diffusion on two scales, that of the pore network and that of the polymeric cell edges. Therefore the shape of water uptake graphs will be more complex than those in Figure 20.2b. Mielewski et al. (1996) measured the diffusivity of water in a solid RIM (reaction injection moulding) PU at 25°C as approximately 4 ⫻ 10⫺11 m2 s⫺1. In Section 20.6.1, Dement’ev et al. (1966) for data water vapour diffusion into thin slices of rigid PU foam is shown to correspond to a diffusivity that is less than 1% of the above value. Therefore, if the PU is in the glassy state, its diffusivity to water will be considerably smaller than when it is in the rubbery state.

20.4

Open-cell foams and sponges 20.4.1 Sponges: natural and synthetic Most of us have an early experience with bath sponges – the structure of soft, natural sponge is similar to that of open-cell foams. They are one of the forimera family, many other members of which contain spiny spicules. Synthetic sponges are often PU foam, coloured to simulate natural sponge. For washing cars, the water must be held until needed, then expelled with a low pressure. PVA foam has a smaller cell size and can be white; it can be used for facial wiping, etc. Shafee and Naguib (2003) described the kinetics of water sorption in crosslinked PVA. The amount of crosslinking agent affects the water uptake (Fig. 20.3), because water causes swelling of the PVA network. Eventually the equilibrium water content is reached. The diffusion of moist air into PVA films was found to be non-Fickian, with an apparent diffusion constant that increased from 0.94 ⫻ 10⫺11 m2s⫺1 at 38% RH, to 1.4 ⫻ 10⫺11 m2s⫺1 at 61% RH, and 2.4 ⫻ 10⫺11 m2 s⫺1 at 74% RH. The hydrogen bonding between the hydroxyl groups in dry PVA is disrupted by the inflow of water. The glass transition temperature of the PVA, initially around 30°C, is reduced somewhat by crosslinking.

20.4.2 Foams used in printer cartridges Polyether PU flexible foams are used in printing rollers, and inside inkjet printer cartridges. Their roles include storage (acting as an ink reservoir) and transport (wicking) of the ink. A patent (Boyd, 1999) 5917527 describes foam inserts, in the ink storage chamber, which regulate the flow of the water-based inks. A negative pressure is required in

486 Polymer Foams Handbook 3 Crosslinking %

h (g water/g polymer)

1% 5% 10% 2

1

0 0

Figure 20.3

200

400 Time (min)

600

800

Crosslinked PVA. The amount of crosslinking agent affects the final equilibrium water uptake (Shafee and Naguib, 2003 © Elsevier).

the chamber, so that the ink does not drool from the micron-sized holes in the print head. The storage chambers for the three colour inks need to be tall and thin, to keep the cartridge compact; this makes the chambers difficult to fill with conventional PU foam. The patent describes a method of ‘felting’ the foam to a compressive strain of 80% at 182°C for 35 min; this permanently compresses the foam from a thickness of 2.3–0.42 in. Thus the firmness (reciprocal of the permanent compressive strain) is 5.5. The small initial cell size (circa 90 pores per inch) provides a high wicking height, and the low initial density of the reticulated foam (21 kg m⫺3) allows a high ink absorption. Similar ‘felted industrial foam’ is described at www.foamex.com. If foam with a relative density 0.03 is compressed to a relative density of 0.3, 70% of the volume is still available for ink storage. If the foam is removed from an exhausted ink cartridge, it expands several times in volume. Figure 20.4 shows how a high wicking height (related to the variable hE in equation (20.1)) can be achieved by using a moderate foam firmness and a liquid contact time exceeding 24 h. Increasing the firmness reduces the foam pore size, and initially increases the wicking height. Thomson (2000) stated that wicking through foams depended on the emulsifier used in preparing the PU foam; if a water-soluble

Chapter 20 The effects of water 175

487

72 48

150

Wick height (mm)

125

24

100 75 50

1 0.5

25 0

4

6

8

10

12

Felt firmness

Figure 20.4

Oil wicking height as a function of the foam firmness and water contact time (h) (redrawn from Foamex).

surfactant was used, the wicking was fast. This suggests that surface layers on the foam control wicking.

20.4.3 Wound care products When foams are used in wound care products, they are expected both to absorb liquid and allow adequate air flow to the surface of the wound. Most such products have several layers. The cut surface of a foam with a large cell size is relatively rough; this increases the adhesion of cells which grow into the foam. It is preferred to have small cell size on the surface in contact with the wound. It is possible to make foams with a porous but smooth skin, in the form required by the application. Hydrophilic PU foams are made in small batches, with typical mould size being a few cubic feet (Thomson, 2000). The pre-polymer and aqueous phases are passed through a mechanical emulsifier, to reduce the size of the pre-polymer spheres suspended in water. The prepolymer is partly crosslinked, so that, once the main reactions start, gelation occurs before the generated CO2 gas can escape. Good temperature control, essential for the successful production of foam, is easier to achieve in small batches. Since the cured foam contains between 50% and 70% water, this must be removed by a drying stage,


200 Q 100

0 0

10

20

30

Time (min) NaHCO3 (mg)

100 µm (a)

Figure 20.5

0

10

20

30

40

50

60

70

90

(b)

(a) Microstructure and (b) effect of the amount of NaHCO3 (mg) on the swelling kinetics of poly(acrylamide co-acrylic acid) (Chen et al., 1999 © Wiley).

which can involve sucking hot air through the foam. This process is slow since much of the water must diffuse from the polymer phase. Burcholtz (1997) described polymers that absorb large volumes of water. Foam can be formed directly by the polymerisation of high internal phase ratio emulsions (Duke et al., 1998). The initial liquid/liquid phase structure is solidified by the polymerisation. Companies such as Rynel (www.rynel.com) produce hydrophilic foams. Chen et al. (1999) showed the microstructure of a super-porous hydrogel (Fig. 20.5a); this was similar to other low-density open-cell foams. The swelling ratio Q (swollen mass – dry mass)/dry mass) is shown as a function of time for a range of copolymer compositions in Figure 20.5b.

20.4.4 Wipers Cooper (1996) compared the number of contaminating particles left on surfaces wiped with open-cell foams and fabrics containing a disinfectant solution. A laundered, sealed-edge, polyester knit wiper released 1200 fibres per m2 and had a sorptive capacity of 290 ml/m2, whereas a hydrophilic PU foam released 16,000 fibres per m2 and held 2060 ml/m2. The greater water capacity of the foam is due to its lower density (and greater thickness?). The debris count figures are not really comparable, since the size and shape of particles are not


(a)

489

(b) Water

Surface of the sample

Water free core

Absorbed water (c)

Figure 20.6

(d)

MRI images of PU foam samples of side 30 mm, (a) low absorption after 8 h, (b) with some water absorbing tracks (arrowed) after 63 days, (c) high absorption after 8 h, (d) after 3 days (Braun et al., 2003 © Elsevier).

specified. He used a hydrodynamic lubrication formula to estimate the thickness of the water layer left on a wiped surface as 80 µm, for a wiping speed of 0.4 m s⫺1 and a wiper length of 0.1 m. However this was an overestimate; the experimental values ranged from 0.2 µm, when the foam was 20% saturated with water, to 1 µm when it was fully saturated. Foamex (2005) describes how their polyether reticulated PU foams can be used for clean-room wipers.

20.4.5 Rigid open-cell foams Braun et al. (2003) used magnetic resonance imaging (MRI) imaging to follow the ingress of water into 30 mm cubes of four PU foams, of density in the range 200–300 kg m⫺3, used for gaskets in a wet environment. They found major differences between the foams; the ‘best’ was only affected for a few mm at the surface after 67 days, while for the ‘worse’ the water had penetrated halfway to the core after 8 h (Fig. 20.6). The latter appears to have contained open cells; intermediate foams appear to have contained some connected porosity, whereas

490 Polymer Foams Handbook the best foam appears to have been closed cell. No details were given of whether the foams were hydrophilic, and, although some water uptake data was given, no diffusivities were calculated.

20.5

Closed-cell bead foams 20.5.1 Water conditioning of EPS products EPS foam is used for the shock-absorbing liners of many types of helmets. BS 6863 (BSI, 1987) required bicycle helmets to be totally immersed for between 4 and 24 h in water at 15 ⫾ 5°C before impact tests. In the current BS EN 1078 there is water spray conditioning for between 4 and 6 h. The following sections describe the kinetics of the water absorption process, and its effects on the mechanical properties.

20.5.2 The advance of the water front Mills and Kang (1994) immersed EPS foam blocks in black fountainpen ink of viscosity the same as distilled water. Cubes of side 25 mm were removed from helmet liners and mouldings with a hot wire cutter, taking care to preserve the inner and outer surface of the moulding. These cubes were immersed in ink for varying times, part-sectioned with a razor blade, then fracturing to reveal the bead boundaries and the position of the ink front. They found that ink initially penetrated along the channels (Fig. 20.8) between beads. On some fracture surfaces ink spread some distance across the polygonal bead faces. No ink was found in the closed cells below the bead faces. The kinetics of the ink advance from the moulded surface into the foam are shown in Figure 20.7 for a box-lid moulding of 22 kg m⫺3 density. The ink penetrated ⬍1 mm into a 25 mm cube, cut from a cycle helmet moulding, after 6 h. Consequently the inter-bead channel network differs from moulding to moulding.

20.5.3 EPS mass increase on water immersion Figure 20.9 shows the relative mass gain (mass gain/original mass) of EPS samples of size 100 ⫻ 25 ⫻ 25 mm vs. time. There are three stages in the water uptake: the first is the immediate filling of cut foam cells; this water cannot be removed completely by wiping with a tissue. The second stage ends after approximately 20 h when water penetrates to the midplane of the foam block and the mass reaches a plateau value (Fig. 20.9a). There is a near-linear increase in the third


491

Penetration from moulded surface (mm)

14 Centre line

12

10

8

6

4

2

0

0

10

20

30

40

50

Ink immersion time (h)

Figure 20.7

Depth of ink penetration from the surface of 22 kg m⫺3 density box-lid foam as a function of the immersion time (redrawn from Mills and Kang, 1994).

10 mm

Figure 20.8

Section through a cube of 56 kg m⫺3 density box-lid EPS after 6 h immersion in black ink; the moulded surfaces are on the left and right (Mills and Kang, 1994).


0.4 Density 22

Density 22 Relative mass increase

Relative mass gain

0.25 Theory 0.20 0.15 0.10 Density 56 0.05

Fourier number Fo 0.5 1.0

0.0 0 (a)

Figure 20.9

10

20

Immersion time (h)

0.3 Density 56 Density 56

0.2

0.1

0.0 30

0 (b)

50

100

150

200

250

Immersion time (h)

Mass gain of water-immersed EPS box-lid mouldings vs. time: (a) for periods up to 24 h, compared with theory and (b) for periods up to 250 h (Mills and Kang, 1994).

stage, which ends at about 240 h (Fig. 20.9b): the water is probably diffusing from the inter-bead channels into the beads. By fitting the theoretical curve to the experimental data in Figure 20.9a, a Fourier number of 1 corresponds to 20 h. Hence, the diffusion coefficient is equal to 7.2 ⫻ 10⫺10 m2 s⫺1. Most of the 40 wt% equilibrium water content, in foam of density 56 kg m⫺3, is present as liquid in the bead boundary channels, but some must be in the cell walls. Samples, much larger in width and length than their 25 mm thickness, were cut from soft shell cycle helmets so the majority of water ingress is from the moulded surfaces. The maximum relative water gain lay between 3%, for a helmet which passed BS 6863, to 8%, for a liner which fractured in BS impacts. In the first material the water did not appear to have reached the midplane of the helmet liner in 250 h, but in the second it had. The total bead channel volume is lower for the helmet mouldings than for the box-lid mouldings. Dusˇkov (1997) found that it took more than 1 year for 100 mm diameter and 200 mm height EPS cylinders of density 20 kg m⫺3 to approach equilibrium water content (Fig. 20.10). The samples were cut from a large blocks of EPS with a hot wire cutter, which may have partly sealed the surface, reducing the water uptake rate. The change in the slope of the water uptake vs. log time was attributed to a change in mechanism, from the initial diffusion into cells on the bead exterior, to a slower diffusion into internal cells. Dusˇkov found that the water uptake did not reduce the Young’s modulus or the compressive collapse stress. He made the unlikely suggestion that water vapour diffusion into the cells increased the total gas pressure, thereby increasing the Young’s modulus. Gnip et al. (2006) measured water uptake in EPS boards of 50 mm thickness when placed 50 mm below the water surface in a tank at


493

Water absorption by volume (%)

2

Maximum 1.5 Mean

1

Minimum

0.5

0 1

10

100

Exposure time (days)

Figure 20.10 Water uptake vs. log time for EPS cylinders of density 20 kg m⫺3 (redrawn from Dusˇkov, 1997 © Elsevier). 1.8 1.5 9

1.2

mw =

wτ w28

7 4 5

0.9

8

6

3

0.6

2 1

0.3 0

0

50

100

150

200

250

Time τ (days)

Figure 20.11 Water uptake vs. time for EPS boards of a range of densities, normalised in terms

of the water uptake at 28 days.The dashed line is a ␶0.25 relationship. (Gnip et al., 2006 © Elsevier).

23°C. The densities ranged from 12 to 35 kg m⫺3. They found a smooth increase in the water absorption with time (Fig. 20.11). Therefore the change in slope in Dusˇkov’s data after 30 days may be an artefact.


Compressive yield stress (at a strain of 10%) as a function of water immersion time yield stress (MPa) Density (kg m⫺3)

Source

21 50 56 70

Box lid Helmet liner Box lid Helmet liner

dry

1 day

24 days

0.125 0.435 0.56 1.07*

0.118 0.368 0.48 0.764*

0.16 0.345

*At the 12% yield strain.

20.5.4 The effect of water on the strength of EPS Table 20.1 shows that the uniaxial compressive yield stress falls by between 6% and 29% as a result of water immersion for 24 h, with higher values for higher density foams. After 24 days immersion, the yield stress of the low-density box lid has slightly increased, whereas that of the 56 kg m⫺3 density box lid has fallen to 61% of the dry value. Figure 20.12 compares the wet and dry compressive stress–strain curves for the box-lid foam of density 56 kg m⫺3; there is no change in the shape but the stress levels were reduced after 24 h immersion. The reduction in yield stress continues while the water absorption process continues. It is likely that the yield stress of the polystyrene cell faces is reduced by absorbed water (plasticisation). Skinner et al. (1965) noted that cell faces in polystyrene bead foams are highly oriented; this may play some part in the high water absorption level. However, the strength of bulk polystyrene is hardly affected by water. A 3 mm thick solid polystyrene moulding, after 24 h immersion, absorbs less than 0.1% by weight of water. The diffusion constant for water in bulk polystyrene is so low that water equilibration takes much longer than 1 day. When bars, cut from the mouldings, were loaded in 3-point bending, failure was sudden with a loud noise as a crack appeared in the centre of the tensile face of the bar. The crack initiated at the boundaries of the surface beads and propagated across the bar. The maximum tensile stress on the bar surface at the fracture initiation point was calculated using equation (5.27). Table 20.2 shows that these flexural strengths are significantly higher than the compressive yield stresses. The flexural strengths can be considered as the tensile strength of bead boundaries near the moulded surface.

20.5.5 Effect of water on thermal conductivity When water is absorbed into closed-cell foam, it increases the thermal conductivity, making the thermal insulation less effective. It can also


495

1.0

Dry 0.8

1 day

Stress (MPa)

0.6

24 days

0.4

0.2

0.0 0.0

0.2

0.6

0.4 Compressive strain

Figure 20.12 Compressive stress–strain curves of 56 kg m⫺3 density box-lid foam, dry and after water immersion for 1 and 24 days. Three test directions were used for the 24 day samples, and the lowest curve is for compression parallel to the moulded surface (Mills and Kang, 1994).

Table 20.2

Flexural strengths as a function of water immersion time Flexural strength (MPa)

Density (kg m⫺3)

Source

Dry

1 day immersion

Dry flexural strength compressive yield stress

22 50 56

Box lid Helmet liner Box lid

0.38 1.75 1.27

0.38 1.35 1.12

3.0 4.0 2.3

reach steel parts, possibly causing corrosion. Dechow and Epstein (1978) found the 0.2 vol.% water absorption of extruded polystyrene (XPS) foam after 5 years to be much smaller than the 8–30% for polystyrene bead foam (EPS).

0.06

14

0.05

12

0.04

10

0.03

8

% vol

+(W/km)


0.02

6

0.01

4

0

2

⫺0.01

0

2

4

6

Measured, sample 1 Measured, sample 2 Calculated 1 Calculated 2

8

0 0

w (vol. %)

(a)

+50/+1°C

(b)

10 20 30 40 Distance from the cold surface (mm)

50

Figure 20.13 (a) Increase in the thermal conductivity of EPS of density 15 (crosses), 20 (trian-

gles) and 33 (circles) kg m⫺3, at mean test temperatures of ⫺5°C (dashed curves) and 10°C (solid curves) as a function of the vol.% water content. (b) Measured and calculated vol.% water distributions through the panel after 28 days exposure with surfaces at 50°C and 1°C (Ojanen and Kokko, 1997 © ASTM).

Ojanen and Kokko (1997) monitored the ingress of moisture into building panels and the effect on the thermal conductivity. The absorbed moisture, with high thermal conductivity, increased the thermal conductivity of EPS insulation (Fig. 20.13a). The water profile in the foam is shown in Figure 20.13b to be almost linear after 28 days exposure to 50°C on one side of the panel and 1°C on the other. In simulations of foam used for frost protection, where the temperature differential across the foam was 5°C, less than 1% water was predicted to be absorbed after 2 years.

20.6

Closed-cell foams 20.6.1 Water absorbed in rigid, closed-cell PU foam Gunn et al. (1974) used a microbalance to monitor water vapour uptake rates in a closed-cell PU foam. However they gave no information about the chemistry or modulus of the foam or the sample dimensions. Although the data was fitted to theoretical mass vs. time graphs, only the relative change in diffusivity with foam density was given. A slow increase in mass of a 30 kg m⫺3 density foam was observed after the diffusion process had saturated; this was attributed to a reversible first-order reaction with the urea content of the foam. Dement’ev et al. (1996) studied a closed-cell rigid PU foam of density 94 kg m⫺3, of a type used for thermal insulation. By using 2 mm thick slices from foams with a mean cell diameter of 3 mm, the test samples were effectively open cell. Water vapour uptake data, at a


Weight gain (%)

3

497

0.97

2 0.69

0.41

1

0.19 0.13 0.06

0

0

20 40 60 Square root of time (s0.5)

Figure 20.14 Water vapour diffusion into PU foam at the marked relative humidities (redrawn from Dement’ev et al., 1996 © Pergamon Press).

range of relative humidity, were plotted against the square root of time (Fig. 20.14). This allowed the water diffusivity at 23°C of the PU to be determined as 1.6 ⫻ 10⫺13 m2 s⫺1. Barker et al. (1999) found that a temperature gradient of about 50°C across a rigid PU foam board causes a higher water penetration than in an isothermal test. The water-barrier skin of manufactured foam board was removed, and samples of 50 mm thickness exposed to air with 75% RH at 50°C. The sample weight increased linearly with time, with a 212% weight increase after 28 days. The thermal conductivity also increases linearly with the water content, from 0.0263 W m⫺1 K⫺1 at zero water content, to 0.0293 W m⫺1 K⫺1 at 120 wt% water content. With the water-barrier skin in place, the water resistance was approximately doubled, while, if an aluminium or low-density polyethylene (LDPE) skin was used, there was negligible change in the water content with time. They associated the wide variation of literature data with variable foam quality and differing test methods. It is possible that some PU foams, rapidly affected by water, contain linked porosity of the type found by Braun et al. (2003) (Section 20.4.5).

20.6.2 Syntactic foams If low-density closed-cell foams are used deep in the sea, the high hydrostatic pressures either compress the foam, or fracture the cell


Buoyancy loss (%)

75

50

Without surfactant Atsurf 3315 Cresmer B246M Tegostab B8404

25

0

0

0.5

1

1.5 2 Water pressure (MPa)

2.5

3

Figure 20.15 Buoyancy vs. water pressure for rigid PU foam (Mondal and Khakhar, 2004 © IUPAC).

faces, so the foam loses its buoyancy. Consequently syntactic foams are used for buoyancy at depth. These contain hollow glass microspheres in a polymer matrix, and have a density less than that of water (Kim and Plubrai, 2004). When closed-cell rigid PU foams are subjected to high water pressures, the cell faces fail and the water enters the structure. Mondal and Khakhar (2004) showed the pressure vs. loss of buoyancy graph (Fig. 20.15) for a foam of density circa 150 kg m⫺3 was a function of the surfactant used in the foaming process, hence of the thickness (strength) of the cell faces. In a subsequent article (2006) they modelled the breakdown process and showed that the threshold pressure for hydraulic collapse occurred when 9% of the cell faces had fractured.

20.6.3 Buoyancy aids for swimming Pool floats and games use closed-cell foams, which need to be buoyant and flexible. These vary in nature from polyethylene (PE) foams, which may not have a very long outdoor life (due to UV degradation of the polymer) to plasticised PVC foams. Extruded cylindrical ‘pool noodles’ are easy to manufacture, and can be readily bent, or sewn into cloth to make support frames. As the hydraulic pressures are minimal, low-density closed-cell foams can be used.


Table 20.3

499

Buoyancy values required EN

Use

393 395 396 399

Competent swimmer near bank/shore Waiting in calm water for rescue General offshore rough weather use As above if carrying significant weights or with clothing which traps air

Buoyancy N 50 100 150 275

20.6.4 Lifejackets The buoyancy aids for life preservation require a combination of mechanical and physical properties. Standards do not prescribe the foam density but rely on performance tests. BS 3595 required a compressive creep test on a specimen of 200 mm by 200 mm by the thickness used; when a creep stress of 88 ⫾ 1 kPa was applied for 1 h, and the foam allowed to recover for 5 min, its buoyancy must not have reduced by more than 7.5%. This test does not simulate the storage of lifejackets under heavy objects for weeks or months, hence it is important that lifejackets are not stored incorrectly. Chapter 19 shows that creep, caused by air loss from the foam cells, is a slow process due to the low diffusivity of air through the foam. No subsequent standard has a compressive creep test. The current standard BS EN 393: 1993 to 399: 1993 for lifejackets and buoyancy aids specify the amount of buoyancy required (Table 20.3). The compressibility tests differ from those in BS 3595. A uniaxial stress of 500 kPa is applied 5 times underwater, the foam is dried and 500 cycles of loading applied in air, then four further underwater compressions applied. The compressive deformation is applied at a rate of 200 mm/min, so the foam is not under a high stress for any significant time (i.e. creep by gas diffusion loss will be negligible). The loss in buoyancy should be less than 10%. The thermal stability test involves a cycle of 7 h at 60°C, 17 h at 23°C, 7 h at 30°C, and 17 h at 23°C. After 10 such cycles the buoyancy loss should not exceed 5%. Zotefoams found that a number of their crosslinked polyolefin foams, with densities ranging from 24 kg m⫺3 (LDPE) to 50 kg m⫺3 (EVA, ethylene–vinyl acetate) meet the EN standard. The compression tests, should really be carried out at 60°C, since the loss of air from foams is most severe at the highest temperatures. The challenge with lifejacket design is to provide buoyancy in places that will keep the wearer floating the right way up, even if unconscious. The lifejacket must not unduly hinder the activities of

500 Polymer Foams Handbook the wearer prior to his/her immersion, yet it must be sufficiently well attached so that it does not displace significantly when the wearer jumps into the water.

References Barker M., Sing S.N. & van der Sande K. (1999) Water vapour condensation resistance of rigid polyurethane foam, Polyurethanes Expo 99, American Plastics Council. Boyd P.V., Huth A.M.C. et al. (1999) Ink-jet pen with near net size porous member, US Patent 5,917,527. Braun J., Klein M.O. et al. (2003) Non-destructive, 3D monitoring of water absorption in PU foams using MRI, Poly. Test. 22, 761–767. BS 6863 (1987) Pedal Cyclists’ Helmets, British Standards Institution, London. Burcholtz F.L., Ed. (1997) Modern Superabsorbent Polymer Technology, Wiley, New York. Chen J., Park H. & Park K. (1999) Synthesis of superporous hydrogels: hydrogels with fast swelling and superabsorbent properties, J. Biomed. Res. 44, 53–62. Cooper D.W. (1996) Comparing foam and fabric wipers for applying disinfectant, Pharm. Tech. February, available at www. itwtexwipe.com. Dechow F.J. & Epstein K.A. (1978) ASTM STP 660, Thermal Measurements of Insulation, American Society for Testing and Materials, Philadelphia, PA, p. 234. Dement’ev A.G., Khlystalova T.K. & Miklheyeva I.I. (1996) Diffusion and sorption of water vapour in polyurethane foam, Polym. Sci. USSR 31, 2291–2296. Duke J.R., Hoisington M.A. et al. (1998) High temperature properties of poly(styrene-co-alkylmaleamide) foams prepared by high internal phase emulsion polymerisation, Polymer 39, 4369–4378. Dusˇkov M. (1997) Materials research on EPS 20 and EPS 15 under representative conditions in pavement structures, Geotext. Geomembr. 15, 147–181. Foamex (2005) Literature on www.foamex.com/technical. Gnip I.Y., Kersulis V. et al. (2006) Water absorption of expanded polystyrene boards, Polym. Test. 25, 635–641. Gomes C.M., Adao M.H. et al. (1997) Wettability of cellular polyurethane, J. Polym. Sci. B Phys. 35, 407–414. Gunn D.J., Moores D.R. et al. (1974) Sorption of water vapour onto polyurethane foams, Chem. Eng. Sci. 29, 549–559. Kim H.S. & Plubrai P. (2004) Manufacturing and failure mechanisms of syntactic foams under compression, Composites A 35, 1009–1015.


501

Luikov A.V. (1968) Analytical Heat Diffusion Theory, Academic Press, London. Mielewski D.F., Anturkar N.R. & Bauer D.R. (1996) Estimation of diffusion and solubility coefficients for water and CO2 in reaction injection moulded parts, Polym. Composit. 17, 649–655. Mills N.J. & Kang P. (1994) The effect of water immersion on the fracture toughness of polystyrene foam used in soft shell cycle helmets, J. Cell. Plast. 30, 196–222. Mondal P. & Khakhar D.V. (2004) Regulation of cell structure in water blown rigid polyurethane foam, Macromol. Symp. 216, 241–254. Mondal P. & Khakhar D.V. (2006) Simulation of the percolation of water into rigid polyurethane foams at applied hydrostatic pressure, Polym. Eng. Sci. 46, 970–983. Ojanen T. & Kokko E. (1997) Moisture performance analysis of EPS frost insulation, STP 1320 Insulation Materials, Testing and Applications, ASTM, West Conshohocken, PA, pp. 442–455. Sabbahi A. & Vergnaud J.M. (1991) Absorption of water at 100°C by polyurethane foam, Eur. Polym. J. 27, 845–850. Sabbahi A. & Vergnaud J.M. (1993) Absorption by water by polyurethane foam: modelling and experiments, Eur. Polym. J. 29, 1243–1246. Saskatchewan University (2003) At interactive.usask.ca/ski/ agriculture/soils/soilphys/soilphys _wat.html. Shafee E.E. & Naguib H.F. (2003) Water sorption in crosslinked poly(vinyl alcohol) networks, Polymer 44, 1647–1653. Skinner S.J., Baxter S. et al. (1965) How polystyrene foam expands, Plast. Eng. 42, 171–178. Thomson T. (2000) Design and Applications of Hydrophilic Polyurethanes, Technomic, Lancaster, PA.

Chapter 21

Rugby and soccer protection case study

Chapter contents 21.1 Introduction 21.2 Soccer shin guards 21.3 Soccer ankle protection 21.4 Rugby goal padding 21.5 Protective headguards for rugby References

504 504 514 520 526 527


21.1

Introduction Foams are used in a number of body protection products used in soccer (Association Football) and rugby, which are worn to prevent injury from impacts with other players, goal posts, or with the ball. To bring some order to the apparently disconnected topics, the geometries of the body part and the object impacted have been classified in Table 21.1; for example, head-to-head clashes are considered as sphere-to-sphere contacts. One example of each type is considered in this chapter. The impact geometry should suggest related products. The foams in these products may be open or closed cell, and only some designs use an outer load-spreading shell. For a given thickness of protective foam, the geometry and the presence/absence of a shell affect the impact response. Finite element analysis (FEA) allows the stress distribution in the foam to be predicted, and the protector response to be calculated.

21.2

Soccer shin guards Shin guards are worn over the front of the tibia where soft tissue coverage is low. The small radius of curvature of the bone means there is small contact area with impacting flat or convex object. This and the bone’s high stiffness means that impact forces and pressures can be high, possibly causing injuries. Some stages in the design process are set out in the next six sections, with the emphasis on testing and FEA. Existing products are often tested and evaluated with players.

21.2.1 Threat evaluation Research discussed in Section 1.6 considered the risk from leg-to-leg contact, but shin guards are unable to absorb high impact energies, so prevent tibia fracture in some tackles. Ankrah and Mills (2004) considered the risk to players’ shins from projections on the sole of the opponent’s boot – aluminium and nylon studs, and thermoplastic polyurethane (PU) blades (Fig. 21.1).

Table 21.1

Impact geometries considered Case

Geometry of impact

Shell

Standard

Stud on shin (soccer) Stud on ankle (soccer) Head on goal post (rugby) Head to head (rugby)

Stud to cylinder Stud to spherical cap Sphere to cylinder Sphere to sphere

Yes Yes No No

BS EN 13061 None None IRB rule

Chapter 21 Rugby and soccer protection case study

505

21.2.2 Injury biomechanics Hawkins and Fuller (1999) found 20% of soccer injuries were contusions and 13% occur to the lower leg. Boden (1998) reviewed leg injury studies and noted that the majority of 31 fractured legs, from a direct blow, occurred while wearing shin guards. Biomechanical criteria for injury differ depending on the part of the body impacted. The criterion for bone fracture, from impact bend tests on cadaver tibias, was a peak force that ranged from 4 to 7 kN (Nyquist et al., 1985), and was 2.9 ⫾ 0.4 kN (Francisco et al., 2000). If the foot is planted on the ground, and the opposing player’s foot loads the tibia near its centre, the kinetic energy of the tackle can exceed the fracture energy of the tibia. Even if a shin guard could shunt the loads to the knee and ankle, this would increase the risk of knee injuries, which are more difficult to treat. The criterion for soft tissue contusions in humans is not established. The anterior border and the medial surface of the tibia have very little soft tissue cover (Fig. 21.2). Crisco et al. (1996) impacted the leg muscle of rats with a 6.4 mm diameter nylon hemisphere to cause contusions; the average pressure over the projected area of the hemisphere reached 9 MPa. Beiner and Jokl (2001) could not decide whether the muscle contusion criterion should involve force, pressure, or another mechanical variable. It is assumed that soft tissue bruising depends on the peak pressure experienced. As a working hypothesis, contusions are expected if the pressure order of magnitude exceeds 1 MPa. Immediate treatment (ice, pressure) can reduce the development of bruising.

21.2.3 Materials selection This is considered separately for the two main components.

12.4 mm

30 mm

12.5 mm

4 mm 14 mm

15.8 mm

(a)

Figure 21.1

12.3 mm

(b)

(c)

(a) Aluminium stud, (b) nylon stud, and (c) PU blades from football boots (Ankrah and Mills, 2004).


Tibia

Figure 21.2

Fibula

Cross-section of the lower leg (Ellis et al., 1994 © Butterworth).

21.2.3.1

External shells

These distribute forces away from bony protuberances, thereby avoiding excessive contact pressures. They may also increase the volume of foam that is compressed. Francisco et al. (2000) commented that fibreglass shin guard shells were better than other materials in distributing the impact force. Thermoplastic shells are typically 2 mm thick. Ankrah and Mills (2003) found shells had different bending stiffness EI in the traverse and vertical directions (E is Young’s modulus and I the second moment of area of the cross-section). There was considerable variation between designs. Some have longitudinal slots to reduce the transverse bending stiffness, while longitudinally segmented ones have a near-zero transverse stiffness, so that the sock tension can bend them to the leg shape (Fig. 21.3). The Umbro design, seen in Figure 21.6, has traverse ridges to increase the bending stiffness. It also has a smaller radius of curvature than the front of the leg, so that the shell stands proud of the tibia. 21.2.3.2

Energy-absorbing foam

Francisco et al. (2000) commented that increasing the compliance, using air bladders, attenuated peak forces, while increased foam thickness


Figure 21.3

507

Cross-section of Adidas segmented guard: the thin PE foam is sewn to cloth, while the tubular PE mouldings are in pockets.

was more important than increased guard length. Typical guards contain a uniform 3–5 mm thickness of low density (⬍100 kg m⫺3), closedcell, ethylene–vinyl acetate (EVA) foam, with occasionally a 5 mm layer of soft, open-cell, PU foam. An Umbro guard had a gap between the shell and the skin over the tibia, partly filled with a thin layer of Confor 47 slow-recovery foam. Davies and Mills (1999) showed that this foam absorbs significant amounts of energy at impact rates, while conforming to the body shape when loaded slowly. However, it is very temperature dependent near the Tg of approximately 20°C. More semi-rigid foam could be inserted into this gap, so long as a soft EVA foam is in contact with the leg.

21.2.4 Wearability and compliance rates Shin guards are the only protective equipment required by FIFA; in professional soccer games the compliance rate is 100%, but in training or amateur soccer it is less. It seems that some players modify guards to make them more comfortable or lighter. Removal of ankle protectors is common practice (Ankrah, 2003). Regulations may need to be introduced that shin guards should (a) cover a minimum area of the shin, (b) contain ankle protection, and (c) not be altered. Their size must not inhibit performance and the foam lining must not cause discomfort. Consequently designs must consider the guard mass, its effect on body ventilation, and whether it restricts mobility.

21.2.5 Product effectiveness Boden (1998) could find no information on reductions in soft tissue injuries since shin guards became mandatory. Forensic investigations of the impact are impossible with EVA and low density polyethylene (LDPE) foams, as they recover their dimensions completely after an impact.

508 Polymer Foams Handbook 21.2.6 Standards and test rigs Product standards are usually established after the product effectiveness is established. However, for shin guards, a standard was developed first. BS EN 13061 (2001) contains two types of impact test. In one a stud, of diameter 10 mm attached to a 1 kg mass, makes an oblique impact at 5.4 m s⫺1 onto a guard, which is supported on a 5 kg metal legform. Since the angle of incidence on the guard surface is close to 45°, the velocity component normal to the surface is 3.7 m s⫺1. From equation (21.3), the effective impact energy is 5/6 of the kinetic energy of the striker, or 5.8 J. The guard must not tear or perforate, but there is no force measurement. The second test uses a flat metal bar, 14 mm wide with 2 mm radius edges, oriented to be across the guard, which is supported on a rigid legform. For an impact with 2 J kinetic energy, the peak force allowed is 2 kN. These test energies determine the product design. Here the low energies probably correspond to the performance of pre-existing products. However, as the next section shows the test energies should be much higher. Unpublished FEA showed that the flat bar test can be passed by a simple guard, having a 2 mm polypropylene (PP) shell over 5 mm of EVA foam. However, the 2 J impact causes the foam to bottom out, so the peak pressure on the leg is 1.5 MPa, which will probably cause bruising. The test rig mechanics should correspond with leg and foot biomechanics, in order to rank products correctly. Given the complex geometry of the foot, its ability to articulate, and the soft tissue components, some simplification is necessary. A poorly designed test rig can produce false product rankings. In particular, the replacement of soft muscle by steel reduces the importance of load spreading in the shin guard. Most research test rigs use rigid strikers; Lees and Cooper (1995) used a 70 mm diameter hemisphere (!) to represent a foot, while Bir et al. (1995) used a metal cylinder of radius 38 mm to represent another player’s leg. However, Francisco et al. (2000) used a 12.7 mm thick rubber cover on the cylindrical ‘leg’ striker. Bir et al. mounted the guards on the lower leg of a Hybrid III car-crash dummy. This is not biomechanically realistic; the polyvinylchloride (PVC) plastisol skin is only intended to hide the stiff central steel rod. In Francisco et al.’s rig, rubber-covered foam (no details given) surrounded a tibia with the correct bending stiffness, simply supported at both ends. Bir et al., who adjusted the striker kinetic energy (not quoted) to give a peak force of 2.3 kN with the unguarded leg, found that guards reduced the peak force by 40–70%. However, Francisco et al., who used test energies of 8–21 J, found reductions of 11–17%. The force reduction values are inconsistent because the test conditions differ.

Chapter 21 Rugby and soccer protection case study 21.2.6.1

509

Effective mass and effective impact energy

The test rig should produce a similar impact force–time trace, and similar pressure distribution on the guard, as that experienced in soccer. However, the striker is usually a single ‘rigid’ mass rather than the connected bony masses and soft tissues of the foot. Some materials properties may be rate dependent; if so, the impact velocity V should be typical of football impacts. If the impact force vs. time trace has a single, short-duration peak, it is possible to model this as a collision between two elastic bodies. The test rig foot mass should be the effective mass Me of the human foot, defined as the mass of an elastic body, with the same momentum as the real foot, which produces the same peak force in an impact. Mills and Zhang’s (1989) analysis, of the impact between an elastic body of mass Me and velocity V with a fixed body (of effectively infinite mass), gave the initial peak force as F ⫽ V Me k

(21.1)

where k is contact stiffness (N m⫺1) between the bodies. Consequently, if the contact stiffness for a kick to a footballer’s leg is known, the value of Me can be calculated from the measured peak force. However, no one has instrumented footballer’s legs and recorded impact peak traces. If, however, the force vs. time trace has a long duration with several peaks (as in the vertical component of the foot strike force in running), a more complex model is required, with several linked masses and springs. It is easier to analyse a guard’s performance, if it is supported on a rigid, immovable anvil. However this tends to increase the impact severity, compared with a real leg, which will move during impact. Gilchrist and Mills (1996) calculated the effective striker kinetic energy in pre-1985 motorcycle helmet standards, in which both the headform and striker moved during impact. Applying this to soccer, we consider a striker, of mass m1 and initial velocity V1, impacting a guarded leg of mass m2 and initial velocity V2 ⫽ 0. As momentum is conserved in the collision, the common velocity Vc of the masses at the moment of nearest approach is Vc ⫽

m1V1 ⫹ m2V2 m1 ⫹ m2

(21.2)

The effective impact energy Ee is defined as the energy input to the protector up until the time when m1 and m2 have a common velocity

510 Polymer Foams Handbook (when there is peak protector deformation). Irrespective of the coefficient of restitution of the protector ⎛ m ⎞⎟ m V 2 2 ⎟⎟ 1 1 Ee ⫽ ⎜⎜⎜ ⎝⎜ m1 ⫹ m2 ⎟⎠ 2

(21.3)

For a fixed-leg test rig, the striker kinetic energy should be the effective impact energy of a football kick. Lees and Nolan (1998) quote peak toe velocities of 16 m s⫺1 in a placed-ball kick. Clauser et al. (1969) give the foot mass as 1.5% of the total body mass, say 1.05 kg for a 70 kg player. Clarys and Marfell-Jones (1994) give the bony mass of the foot as 31% of the total, say 0.32 kg for the same player. However, foot flexibility is considered in modelling foot strike forces in running (Giddings et al., 2000). If this approach is used, the effective mass of the forefoot bones may be 艑0.1 kg; for a 16 ms ⫺1 velocity the kinetic energy is 13 J. If a forefoot strikes a lower leg bone of mass 0.62 kg, the effective kinetic energy by equation (21.3) is 11 J. 21.2.6.2

Direct stud impacts on shin guards

Ankrah and Mills (2003) assessed shin guard protection against studs, under a worse-case scenario, with the stud aligned with the centre of the tibia, and a direct rather than oblique impact. The justification is that the direct component of the impact velocity causes the bruising injury, while the tangential velocity component only causes the guard to slide on the leg, and does not increase the pressure on the leg. If the guard shell is not penetrated, there is unlikely to be skin penetration. A glass-fibre reinforced epoxy tibia with a foam core (www. sawbones.com) had a biofidelic bending stiffness of 180 Nm2. It was supported at both ends and placed at the front of a 100 mm diameter cylinder of ‘Senflex 435’ ethylene–styrene interpolymer (ESI) foam that simulated muscle (Fig. 21.4). A 2 mm thick silicone rubber cosmetic skin held the components together. The striker was fitted with a linear accelerometer, aligned vertically. Impact energies of ⬍5 J were used, to avoid damage to the rig, rather than the 11 J calculated in the last section. The striker force was calculated from the product of striker acceleration and mass. Numerical integration of the acceleration signal produced a graph of the striker force vs. the deflection of the upper surface of the guard at the point of impact. Figure 21.5 compares such graphs for a number of shin guards; the initial low slope is due to the compression of soft PU and EVA foams. Once the foams have bottomed out at 5–7 mm deflection, the higher slope is due to the stiffness of the tibia in 3-point bending.


Figure 21.4

511

A shin guard on Ankrah and Mills (2003) test rig. The shadow of the tibia (arrowed) is visible through the silicone rubber skin. 1.4 Adidas basic Nike

1.2

Stud force (kN)

1.0 0.8 0.6 0.4 Adidas segmented

FEA 0.2 0.0

0

1

2

3

4

5

6

7

8

9

10

11

12

13

Deflection (mm)

Figure 21.5

Experimental stud impacts with 5 J energy onto three types of shin guard, compared with the FEA prediction for a 2 mm PP shell over 5 mm of low density EVA foam (Ankrah and Mills, 2003).

21.2.6.3

Load spreading and the shell resistance to buckling

Ankrah and Mills (2003) mounted Tekscan Flexiforce sensors of load range 111 N on the rubber skin, with three along the tibia and four on the ‘soft tissue’ (Fig. 21.6c). The resistance of these flexible 9.5 mm diameter sensors changes with the applied compressive force. The stud is aligned with the sensor 2. The peak striker force (Table 21.2) reached 1100 N for 3.7 J impacts. The peak local force of 86.7 N, on the tibia under the stud, corresponds with a peak pressure of 1.21 MPa. If bruising occurs for pressures ⬎1 MPa, some shin guards allow bruising for a 4 J stud impact over the tibial crest. The shells have buckled and


6 cm 7

3 cm 2 5

4 1

(a)

Figure 21.6

Table 21.2

(b)

(c)

Frames of peak shell deformation in a stud impact: (a) Adidas basic, (b) Umbro, and (c) pressure sensor locations, relative to the tibia (shaded band) (Ankrah and Mills, 2003).

Maximum forces for shin guards, impacted by a metal stud with 3.7 J energy Shin guard

Striker force (N)

Sensor 2 force (N)

Slong

Strans

1231 1066 867 846 778 622 1096 550

80.9 78.8 86.7 86.7 72.9 36.0 86.7 24.6

0.014 0.075 0.072 0.161 0.073 0.497 0.044 0.758

0.005 0.006 0.007 0.020 0.031 0.121 0.007 0.038

None (PU blade) Adidas basic Adidas segmented Adidas shelter Adidas venom Grays Nike Umbro

the foams bottomed out at this stage, so the pressure will rise rapidly for higher impact energies. The measures for load spreading along the leg Slong and in the transverse direction Strans were Slong ⫽

F1 ⫹ F3 2F2

Strans ⫽ 0.75

F4 ⫹ F5 ⫹ F6 ⫹ F7 F1 ⫹ F2 ⫹ F3

(21.4)

The scale runs from 0 for no load spreading to 1 for uniform pressure. The shell bending stiffness in the longitudinal [EI]long and transverse directions [EI]trans was measured with a 50 N load. The correlation coefficient between Slong and [EI]long is r2 ⫽ 0.84, and that between Strans and [EI]trans is r2 ⫽ 0.63, showing that increased shell bending stiffness improved load spreading from a central load. The guards with the largest S values (in bold in Table 21.2) had the lowest peak forces on sensor 2 under the stud. It is impractical for a guard to have a greater longitudinal bending stiffness than the tibia (180 Nm2). However, it


513

is easier to spread load along the tibia than to its sides. The Umbro thermoplastic shell, with the highest transverse bending stiffness, only achieved Strans ⫽ 0.04, while most guards have much lower values. High-speed photographs (Fig. 21.6) showed that most shells buckled; the Adidas basic buckled by a load of 696 N, while the Umbro Armadillo maintained its shape at a load of 583 N. 21.2.6.4

FEA of direct stud impacts on guards

Soft tissue was modelled as a nearly incompressible, gel-like material, using a hyperelastic model with shear moduli µ1 ⫽ µ2 ⫽ 200 kPa, exponents α1 ⫽ 2 and α2 ⫽ ⫺2, and inverse bulk modulus D ⫽ 1.0 ⫻ 10⫺9 Pa⫺1 (Verdejo and Mills, 2002). The tibia was modelled as a hollow tube of constant cross-section, with shape shown in Figure 21.7, and bending stiffness 180 Nm2. It was placed symmetrically in the model (Fig. 21.2 shows it is asymmetric in the human leg) to cut the problem size. The external shape of the leg muscle was a cylinder of diameter 100 mm; it was bonded to the tibia, with zero muscle cover at the apex. The tibia is simply supported at the ankle and knee. The guard foam was bonded to the shell while the coefficient of friction between the foam and the muscle was 0.75. To reduce computation time, two planes of mirror symmetry were used in ABAQUS 6.2 standard. The predicted force vs. deflection relationship (Fig. 21.5) for a shell of modulus 1 GPa has approximately the slope of the initial part of the experimental data. The simulation ceases once the foam has bottomed out under the stud, whereas the experimental response continues with the tibia bending. Later, using dynamic FEA, the slope was predicted to increase after 5 mm deflection. In Figure 21.7a, a low bending stiffness shell, typical of commercial guards, has buckled inwards due to the localised pressure from the stud, causing high foam compressive strains (and stresses) near the stud. Much of the sides of the shell have moved away from the muscle. To prevent buckling (Fig. 21.7b) the shell bending stiffness must be greater than in any commercial shell; there is a greater area of contact between the foam and the sides of the leg muscle, the strain in the foam is less localised. Hence more energy is absorbed before the foam compressive stress near the stud exceeds about 300 kPa, at the maximum deflection of the simulation.

21.2.7 Discussion It would be useful to relate football players’ contusion patterns to the type of shin guards worn. At present materials selection is hindered by the uncertain criterion for muscle bruising. If it occurs for pressures ⬎2 MPa, the foam chosen must compress to high strains for stresses

514 Polymer Foams Handbook 3 mm PP shell 5 mm EVA foam

Horizontal mirror plane

Muscle 100 kPa

50 kPa

Stud

1

227 N

Vertical mirror plane Tibia

3 2

(a) 3 mm GRP shell 10 mm EVA foam

Muscle 50 100 150 1

Horizontal mirror plane

Stud Tibia 1337 N

3 2

Vertical mirror plane

(b)

Figure 21.7

FEA predictions at 5.0 mm deflection, with contours of compressive stress in the two directions at 50 kPa intervals: (a) 5 mm foam, shell E ⫽ 1 GPa and (b) 10 mm foam, shell E ⫽ 10 GPa (Ankrah and Mills, 2003).

just below 2 MPa. It is possible to increase protection levels, but the design depends on the bruising criterion.

21.3

Soccer ankle protection The shell of an ankle protector is a shallow spherical cap, which covers the surface of a foam layer. It is held in place over the ankle bone


515

Tibia

Fibula Airex foam Leather Malleolus

Calcaneus (a)

Figure 21.8

Aluminium bones (b)

(a) Cast left ankle bones and (b) impact rig (Ankrah, 2002).

(malleolus), usually by being enclosed in a sock-like extension of a shin protector. The buckling resistance of the shell will be investigated, as well matching the foam properties to those of the shell and underlying soft tissue, to optimise the protection. In common with hip protector design (Chapter 17) the aim is to divert impact forces from a protruding bone, here the ankle (Fig. 21.8). The protector shell can shunt forces to other parts of the foot, and the foam can reduce the maximum forces.

21.3.1 Materials 21.3.1.1

The shell

There is an LDPE shell on the ankle protection of some commercial shin guards: a 2 mm thick injection moulded disc, of diameter 50.8 mm, and radius of curvature 40 mm. Ankrah’s (2002) prototype protectors used a domed shell of diameter 60 mm and radius of curvature 38 mm. 21.3.1.2

Foams

The protector is a multi-impact product, and its major energy-absorbing component is the foam. The foam, by compressing, increases the impact duration and reduces the peak forces. Its performance must remain constant after multiple impacts. Its selection is considered here,

516 Polymer Foams Handbook while its thickness is considered through prototype testing and FEA. Two commercial shin guards contained 5 mm thick EVA foam discs of density 36 kg m⫺3. Prototype protectors used closed-cell crosslinked foam (Table 21.3). ‘ESI/PE (polyethylene)’ foam is a 60:40 blend, of an ESI with LDPE. The foam compressive stress–strain curves are shown in Figure 21.9. The data for the ESI foam was fitted by equation (11.4) with parameters σ0 ⫽ 195 kPa and p0 ⫽ 55 kPa (Ankrah, 2002). The currently used EVA foam is comfortable, but sub-optimal for energy absorption, as its response is mainly that of the compressed cell gas. The foam should have a higher initial compressive yield stress and should be thicker. ESI and ESI/PE foams, if of suitably high density, have such a response, yet recover well from impacts (Ankrah et al., 2002). The inside of such protectors would need a layer of soft EVA foam for comfort.

Table 21.3

Materials used in prototype protectors

Material

Abbreviation

Ethylene vinyl acetate foam 40% ESI/60% LDPE foam Ethylene–styrene interpolymer foam Glass fibre prepreg, 55 vol% fibre Carbon fibre prepreg, 55 vol% fibre Polycarbonate

EVA ESI/PE ESI GRP CFRP PC

Density (kg m⫺3)

Young’s modulus (GPa)

30 53 76 1577 1448 1184

35 55 2.4

0.8

Ogden Stress (MPa)

0.6

0.4

ESI

0.2

ESI/PE 0.0

0

20

EVA

40

60

80


Figure 21.9

Compressive impact stress–strain curves at 20°C of EVA, ESI/PE, and ESI foams, compared with Ogden equation fit to the EVA data (Ankrah and Mills, 2004).


517

21.3.2 Ankle test rig The soft tissues around the ankle were characterised for six soccerplaying students. While the inside of the subject’s left ankle was rigidly supported, a quartz force cell, with a 6.3 mm diameter flat end, was pushed into the skin at one of four locations (Fig. 21.10), just outside the malleolus. Straight lines were fitted to the force vs. displacement graphs. The ankle test rig (Fig. 21.8b) had an indentation response that fell in human ankle response corridor. Its loading slope was close to the mean human subject values, at all the measurement positions (Table 21.4). An aluminium alloy replica was cast from a Sawbones plastic model of the left foot bones (Fig. 21.8a). The fibular malleolus is a 20 mm long ridge, with a radius of curvature of 3 mm, on top of a bone of diameter approximately 20 mm. The ‘bones’ were rigidly supported. The surrounding ligaments, tendons, and muscles were simulated by ESI foam. The malleolus was covered with 2 mm layer of leather to simulate skin. 25

Force (N)

20

15

1 2 3 4 5 6 Ankle rig

A D B C

10

5

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Distance (mm)

Figure 21.10 Indentation tests at position A on ankles.The bold line is the test model response, while the insert shows measurement positions (Ankrah and Mills, 2004).

Table 21.4

Loading slopes (N mm⫺1) for ankle tissue indentation position Position Test rig Subject average ⫾ standard deviation

A

B

C

D

8.3 6.4 ⫾ 3.1

3.0 2.0 ⫾ 0.8

2.6 2.3 ⫾ 1.9

2.8 1.4 ⫾ 1.3


(a)

(b)

Figure 21.11 Frames from high-speed film for (a) GRP and (b) LDPE shells with EVA foam, impact energy 1.7 J, before impact and at peak deflection (Ankrah and Mills, 2004).

21.3.3 Impact tests Studs were attached to a falling block of total mass 1.7 kg, which impacted the ankle at approximately 1 m s⫺1 velocity. For high-speed photography, the protectors were supported on an aluminium hemisphere of diameter 37 mm, rather than the ankle model, to make the deformation axially symmetric. The photographs (Fig. 21.11) confirmed the FEA modelling. The stiffer polycarbonate (PC), glass, and carbon fibre shells maintain their shape on impact, while the LDPE shells buckle.

21.3.4 FEA modelling The fibular malleolus geometry was simplified to be a cylinder with an 8 mm radius hemispherical end, so the model had an axis of rotational symmetry. The 2 mm thick protector shell had a radius of curvature of 40 mm and a radius of 27 mm when projected on a flat surface. The stud was a truncated cone with taper angle of 11°, lower diameter 12 mm, and edge radius 1 mm. The ankle tissue was bonded to the


Table 21.5

519

FEA simulations of a direct stud impact on an ankle protector with 10 mm of EVA form for various shell materials

Shell material

Young’s Modulus (GPa)

PC PP HDPE LDPE

2.4 1.0 0.3 0.1

Maximum stable deflection (mm) 7.2 7.0 3.4 9.1

Peak force (N)

Maximum energy (J)

342 287 42 153

0.98 0.82 0.045 0.444

HDPE: high density PE.

‘fibula’, as was the protector foam to the protector shell. A friction coefficient of 0.7 was used between the other surfaces, but no sliding was predicted. The bone and EVA foams were modelled with the parameters used for the shin guard FEA, while the ankle soft tissue was modelled with the leg muscle parameters. The aluminium stud had a Young’s modulus 70 GPa and Poisson’s ratio 0.4, while the shell material parameters are given in Table 21.5. The shell is predicted to flatten near the stud contact, then to buckle inwards; the force remains less than 50 N for deflections up to 5 mm. Figure 21.12a shows a buckled LDPE shell with a high compressive strain in a small central area of foam, while the majority of the foam is undeformed. To avoid buckling, the material Young’s modulus must be at least 1 GPa. In Figure 21.12b, the PC shell maintains its shape, so the foam compresses almost uniformly. The inherently stiff double curvature of the shell means that PP, with 2 GPa Young’s modulus can prevent buckling, whereas a 10 GPa modulus (i.e. fibreglass) is needed in a 3 mm thick single-curvature shell of a shin guard. The compressive peak stress σ22 in the foam is just over 150 kPa for the PC shell, but 250 kPa for the LDPE shell, at 7 mm deflection. The stresses on the surface of the lateral malleolus are higher at about 500 kPa. The FEA become unstable, either when a kink formed in the outer surface of the foam or when the foam strain became high. The slope of the predicted responses (Fig. 21.13) increased with the shell material Young’s modulus. The 10 mm foam thickness in the prototype ankle protectors was twice that in commercial products. Even so, with an LDPE shell that buckled on impact, the maximum pressure of 5 MPa for a 1.7 J impact could cause bruising. As the estimated effective energy from a football kick can be 10 J, such a protector is inadequate. If a PP shell is used, it should not buckle, so will allow a larger volume of foam to be compressed and absorb energy. It also spreads load to a large area of the ankle. Hence simple changes in materials could improve the protection against bony and soft tissue injuries.


Stud LDPE

200

150

100 50 kPa EVA foam

Muscle (a)

Fibula

Stud PC

150

100 EVA foam 100 50 Muscle

(b)

Fibula

Figure 21.12 FEA predictions for ankle protector shape after stud deflection by 7 mm: (a) LDPE shell and (b) PC shell. Contours of stress σ22 in kPa are shown. There is a rotational symmetry axis at the left (Ankrah and Mills, 2004).

Figure 21.14 shows, for 3.5 J blade impacts on ESI/PE foams, the maximum force for the PC shell is 41% that for the LDPE shell. PU blades are softer than studs and distribute the load from the foot better, while the PC shell is sufficiently stiff not to buckle. It appears that changes from studs to blades might be necessary to reduce ankle injuries.

21.4

Rugby goal padding 21.4.1 Introduction Foam padding is used on rugby goal posts, but there are no specifications or test criteria. The International Rugby Board (IRB) (1998) laws state that, ‘when padding is attached to the goal posts, the distance


521

0.40 0.35

PC PP HDPE LDPE No shell

Force on stud (kN)

0.30 0.25 0.20 0.15 0.10 0.05 0.00

0

1

2

3 4 5 6 7 8 Deflection of protector centre (mm)

9

10

Figure 21.13 FEA predictions of force vs. deflection for various shell materials with 10 mm EVA foam (Ankrah and Mills, 2004).

1.2

1.0

Force (kN)

0.8

LDPE

0.6

0.4 PC 0.2

0

0

5

10 15 Distance (mm)

20

25

Figure 21.14 Force vs. distance for blade impacts with 3.5 J energy on PC or LDPE shells with ESI/LDPE foam (Ankrah and Mills, 2004).

from the goal line to the exterior edge of the padding must not exceed 300 mm’. As the static padding can be much thicker and heavier than the soccer products considered, there is no need for a load-spreading shell. As the padding also acts as an advertisement platform, important

522 Polymer Foams Handbook to the finances of the game, its size can be large. A typical cost for a set of four pads is £200. Price competition between manufacturers of the post-padding could lead to the use of foam of lower density and impact resistance.

21.4.2 Risks and injuries Presumably padding was introduced in the aftermath of player injuries, but no literature can be found. Janda et al. (1995) considered goal post-injuries in soccer, considering round and square section goal posts of unspecified diameter. For headform impacts of only 2.5 J kinetic energy, they found average peak forces of 5 kN with unpadded goal posts and 2–3 kN when 20 mm of PE foam was added. Mills and Lyn (2001) considered a worse-case scenario, in which the player’s hit the rugby goal post on its midline. If the impact is offcentre, the player’s head will deflect to one side, and the padding may rotate on the post; the peak head linear acceleration will be less, but there may be some head rotational acceleration. At the estimated maximum impact velocity of 5 m s⫺1 (11 mph), the head kinetic energy is 62.5 J.

21.4.3 Foam Chapter 2 describes PU chip foam. The other foams considered by Mills and Lyn (2001) were a Combustion Modified High Resilience PU foam (CMHR26) of nominal density 26 kg m⫺3, and an extruded PE foam of a type used in building insulation (Table 21.6). The compressive stress–strain curves (Fig. 21.15) for an impact velocity of 4.4 m s⫺1 are similar for the two soft PU foams, with a near-constant stress of about 5 kPa up to 30% strain (the traces are ‘clipped’ at high stresses). The PE foam has an initial collapse stress of about 80 kPa, then hardens with increasing strain.

21.4.4 Finite element modelling The parameters of the hyperfoam model for the PU chip foam are given in Table 10.2. The headform was modelled as a rigid sphere of radius 90 mm. Square foam padding, of outer dimensions of 250 mm by 250 mm, was supported by, but not bonded to, a cylindrical wooden pole of diameter 100 mm and length 1 m, built into rigid supports at both ends. A coefficient of friction of 0.75 was used between the foam and the headform or pole. To reduce the computation time, two mirror


Table 21.6

523

Foams examined Polymer

Manufacturer

Foam density (kg m⫺3)

PU chip foam PU CMHR26 PE

Kay Metzeler British Vita Unknown

62 25 35

200 PU 26

PE 35

Stress (kPa)

160

120 PU chip 62 80

40

0

0

20

40 60 Compressive strain (%)

80

100

Figure 21.15 Uniaxial compressive stress–strain curves for foams (Mills and Lyn, 2001). symmetry planes were used. Figure 21.16 shows the contours of the compressive principal stress are closely bunched near the indenting headform. The maximum stress, in the highly inhomogeneous stress field, is 250 kPa at this stage. The predicted force vs. deflection graph (Fig. 21.17) has a positive curvature. When the foam bottoms out, the head-to-pole contact area is smaller than for an impact with a flat surface.

21.4.5 Experimental impacts An aluminium headform, of circumference 58 cm and mass 4.7 kg, and shape defined by BS EN 960: 1995, was fitted with a linear accelerometer, aligned vertically. Its radius of curvature at the crown impact site is 100 mm for the fore-and-aft section and 75 mm for the ear-to-ear section. Its guided frictionless fall was from heights up to 2.0 m. The fore-and-aft axis of the headform was at 90° to the length of the pole, which was simulated using a thick-walled high density PE pipe of


Horizontal mirror plane 150 100 50 kPa Foam

Pole

Vertical mirror plane

Figure 21.16 Predicted deformed shape of 250 mm2 PU chip foam padding after 70.2 mm deflection by the headform, with contour levels of compressive principal stress (Mills and Lyn, 2001).

Headform force (kN)

1.5

1

0.5

0

0

10

20

30 40 50 Padding deflection (mm)

60

70

Figure 21.17 Predicted headform force vs. padding deflection graph for the simulation of Figure 21.16 (Mills and Lyn, 2001).

diameter 112 mm. At an impact load of 10 kN, its deflection was less than 4 mm, similar to that of a 1.5 m long wooden pole with its end buried in the ground. The foam outer cross-section shape was a square of 250 mm side, except for the PU chip foam, which was cylindrical of 100 mm thickness. The maximum deflection values in Table 21.7 are close to the foam thickness plus the 4 mm pipe deflection. The


Table 21.7

525

Headform impact test results

Foam HR26 PU chip PE

Thickness at impact site (mm)

Drop height (m)

Input energy ( J)


Maximum force (kN)

Safe energy ( J)

75 100 75

0.75 1.75 2.0

35 82 94

73 84 58

8.1 8.2 3.2

⬃20 ⬃50 ⬎100

5 PU 25

4

PU chip 62

Force (kN)

PE 35 3

2

1

0

0

20

40

60

80

100

Impact energy (J)

Figure 21.18 Impact force on headform vs. impact energy for the three types of post-padding (Mills and Lyn, 2001).

HR26 foam appears to have bottomed out, and the PU chip foam is close to doing so. Figure 21.18 shows the experimental headform force vs. the impact energy, calculated as the integral of the force–distance graphs. The soft HR26 foam bottoms out when 10 J energy is absorbed, then the force rises almost linearly as the support pole deforms. For the denser PU chip foam there is an upward curvature to the relationship. The PE foam response is closer to an ideal one. If 100g head acceleration is allowable (a force limit of 5 kN), the energy absorbed safely is given in the last column of Table 21.7. Hence, if the maximum head impact velocity is 5 m s⫺1, and concussion must be prevented, a 100 mm thick layer of PU chip foam of 64 kg m⫺3 density is just about adequate. The FEA underestimates the headform force for the PU chip foam at 50 mm deflection, by about a factor of 3. However the shape of the predicted relationship is correct, and the foam is predicted to bottom

526 Polymer Foams Handbook out at about 15 J energy input; the FEA shows the region of highly compressed foam is small compared with the size of the post-padding. The errors must relate to faults in the modelling parameters, perhaps due to the neglect of viscoelasticity, and the initial low plateau stress of the foam. In ideal goal post-padding, the force on the head would be nearly constant at a level just below that needed to cause concussion. The PE foam response approaches this ideal, yet the majority of post-padding is PU based. Protection levels should be specified for rugby postpadding, and impact tests performed. If soft PU foam is used, concussion could occur for head impact energies of 20 J, corresponding to head velocities of 2.8 m s⫺1 (6 mph); 20 J is little more than the natural protection provided by the scalp and any skull deflection.

21.5

Protective headguards for rugby Wilson (1998) discussed head injury mechanics and the effectiveness of protective headgear in rugby union, without committing himself to specific performance levels. Gerrard (1998) reviewed the use of shoulder padding and shin guards. He refers to the view that shoulder padding can be viewed as an offensive weapon. A similar attitude may have affected the governing body, the IRB. The IRB laws for headguards (1998) specify the foam density must be less than 45 kg m⫺3 with a tolerance of 15 kg m⫺3, while the maximum padding thickness is 10 ⫾ 2 mm. The laws set minimum (200g) as well as maximum (500g) head accelerations for head impact test. The minimum level may be to prevent the use of ‘aggressive’ headgear. However, 200g is likely to cause severe concussion or brain damage, and the maximum level is at variance with those used for other helmets. The impact energy 13.8 J is very low, compared with the NOCSAE standard for American football helmets. The test sites are the crown, forehead, and temple sweatband area. The foam density and thickness restraints make it almost impossible to design headgear that is effective in preventing concussion. McIntosh and McCrory (2000) tested eight soft helmets used in ‘football’ (Australian rules), under conditions close to those of the IRB, with a 0.3 m drop height and a magnesium headform. They found that three of eight designs failed to meet the IRB specification. The best performing of these contained 12.5 mm of LDPE foam (personal communication). They claim that the IRB considers that rugby headgear prevents abrasions and lacerations, and it is not intended to prevent concussion. McIntosh et al. (2004) tested prototypes of improved football headgear (Fig. 21.19a). They found, by dropping an instrumented headform onto a rigid flat plate, that the performance could be considerable improved by using the thicker foams (Fig. 21.19b).


527

Peak headform acceleration (g)

600

(b)

12

400

16

14 300 200 100 0 0.2

(a)

10

500

0.4

0.6

0.8

Drop height (m)

Figure 21.19 (a) Honeycomb headgear and (b) impact performance for varying thicknesses (mm) of LDPE foam of density 60 kg m⫺3 (McIntosh et al., 2004 © BMJ).

In Australia, headbands have been developed for potential use by car occupants. In a series of reports, Anderson et al. (2001) and Ponte et al. (2002) considered the selection of the best foams, and the design of such headbands. Not surprisingly, they selected semi-stiff foams of thickness circa 20 mm to prevent concussion to the car occupants. These designs might well, with better positional stability, be suitable for use in rugby.

References Anderson R., Ponte G. et al. (2001) Further Development of a protective headband for car occupants, Australian. Transport Safety Bureau Report CR 205 on www.atsb.gov.au/roads/rpts/. Ankrah S. (2003) Protective materials for sporting applications – football shin guards, Ph.D. thesis, University of Birmingham, Birmingham. Ankrah S. & Mills N.J. (2003) Performance of football shin guards for direct stud impacts, Sport. Eng. 6, 207–219. Ankrah S. & Mills N.J. (2004) Analysis of ankle protection in association football, Sport. Eng. 7, 41–53. Ankrah S., Verdejo R. & Mills N.J. (2002) The mechanical properties of ESI/LDPE foam blends and sport applications, Cell. Polym. 21, 237–264. Beiner J.M. & Jokl P. (2001) Muscle contusion injuries: a review, J. Am. Assoc. Orthop. Surg. 9, 227–237.

528 Polymer Foams Handbook Bir C.A., Cassata S.J. & Janda D.H. (1995) An analysis and comparison of soccer shin guards, Clin. J. Sport. Med. 5, 95–99. Boden B.P. (1998) Leg injuries and shin guards, Clin. Sport. Med. 17, 769–777. BS EN 960 (1995) Headforms for Use in the Testing of Protective Helmets, British Standards Institution, London. BS EN 13061 (2001) Protective Clothing – Shin Guards for Association Football Players, British Standards Institution, London. Clarys J.P. & Marfell-Jones M.J. (1994) Soft tissue segmentation of the body and fractionation of the upper and lower limbs, Ergonomics 37, 217–229. Clauser C.E., McConville J.T. et al. (1969) Weight, Volume and Centre of Mass of Segments of the Human Body, AMRL-TR-69-70, Wright Patterson Air Force Base, Ohio. Crisco J.J., Hentel K.D. et al. (1996) Maximal contraction lessens impact response in a muscle contusion model, J. Biomech. 29, 1291–1296. Davies O.L. & Mills N.J. (1999) The rate dependence of Confor PU foams, Cell. Polym. 18, 117–136. Ellis H., Logan B. et al. (1994) Human Cross-Sectional Anatomy, Butterworth-Heinemann, Oxford. Francisco A.C., Nightingale R.W. et al. (2000) Comparison of soccer shin guards in preventing tibia fracture, Am. J. Sport. Med. 28, 227–233. Gerrard D.F. (1998) The use of padding in rugby union, Sport. Med. 25, 329–332. Giddings V.L., Beaupre G.A. et al. (2000) Calcaneal loading during walking and running, Med. Sci. Sport. Exerc. 32, 627–634. Gilchrist A. & Mills N.J. (1996) Protection of the side of the head, Accid. Anal. Prev. 28, 525–535. Hawkins R.D. & Fuller C.W. (1999) A prospective study of injuries in four professional football clubs. Br. J. Sport. Med. 33, 196–203. International Rugby Board, Dublin (1998) Web site: www.irfb.com/ laws/Standard performance specification for items of players’ clothing. law 4m-98g. Janda D.H., Bir C.A. et al. (1995) Goal post injuries in soccer, Am. J. Sport. Med. 23, 340–344. Lees A. & Cooper S. (1995) The shock attenuation characteristics of soccer shin guards, in Sport Leisure and Ergonomics, Eds. Atkinson G. & Reilly T., Spon, London, pp. 130–135. Lees A. & Nolan L. (1998) The biomechanics of soccer, J. Sport. Sci. 16, 211–234. McIntosh A.S. & McCrory P. (2000) Impact energy attenuation performance of football headgear, Br. J. Sport. Med. 34, 337–341. McIntosh A., McCrory P. & Finch C.F. (2004) Performance enhanced headgear: a scientific approach to the development of protective headgear, Br. J. Sport. Med. 38, 46–49.


529

Mills N.J. & Lyn G. (2001) Design of foam padding for rugby posts, in Materials and Science in Sports, Eds. Froes F.H. & Haake S.J., TMS, Warrendale, PA, pp. 105–117. Mills N.J. & Zhang P.S. (1989) The effects of contact conditions on impact tests on plastics, J. Mater. Sci. 24, 2099–2109. Nyquist G.W., Cheng R. et al. (1985) Tibia bending: strength and response, SAE Trans. 94, Section 6, 240–253. Ponte G., Anderson R. et al. (2002) Development and testing of prototype of a protective headband for car occupants, Australian Transport Safety Bureau Report CR 210 on www.atsb.gov.au/ roads/rpts/. Verdejo R. & Mills N.J. (2002) Performance of EVA foams in running shoes, in The Engineering of Sport 4, Eds. Ujihashi S. & Haake S.J., Blackwell, Oxford, pp. 580–587. Wilson B.D. (1998) Protective headgear in rugby union, Sport. Med. 25, 333–337.

Index

acoustic properties, 198 adiabatic, 259, 262, 291, 317 aging modifier, 60 air channel, 17, 192, 276, 458 air-flow during compression, 192 air flow measurement, 178 airflow modelling, 193 airflow resistivity, 179 anisotropic cell, 11, 32, 169, 466 ankle protector, 514 FEA, 518 foam, 515 test rig, 517 annealed microstructure, 16 bead boundary, 79, 81, 336, 367, 369 bead boundary channel, 73, 276, 482, 492 bead foam, 17, 69, 275 air flow analysis, 190 density variation, 75 EPP compressive response, 137 EPS permanent strain, 275 fracture, 366 microstructural rules, 80 tensile strength, 99 thermoplastics, 70 water permeability, 484 water uptake, 480, 482, 490 bead fusion, 71, 73, 74, 75, 76, 81 bead shape, 74, 79, 82 bend test, 86, 99, 106, 367 bending stiffness, 151, 431 biaxially oriented LDPE film, 254 biaxially oriented PS film, 43 bicycle helmet components, 330 effectiveness, 348 FEA of oblique impacts, 345 material selection, 335 microshell, 336 standards, 333 ventilation, 337

biomechanics, 308 falls, 407 head injury, 331 running, 313 seating, 206 soccer injuries, 505 blowing agent, 21, 44, 50, 51, 54, 60, 70, 79, 257, 311, 428 brain injury criteria, 331 branched, 49 branched PP, 46, 49, 58 bubble growth, 13, 53, 54 bubble nucleation, 13, 428 buckling, 94, 152, 268 bulk modulus, 265 Kelvin model, 264 Ogden model, 120 buoyancy aid, 498 capillary radius, 481 car seat, 214 compression, 225 creep, 450 foam selection, 222 posture, 206 transmissibility, 216 vibration, 215 cardboard, 283, 286, 301 cell PE foam, 473 cell deformation, 149, 253, 259 cell gas, 56, 143, 252, 254, 255, 268, 269, 324 diffusion, 470 loss, 440, 471, 476 temperature, 317 cell growth, 13 chip foam, see remoulded foam rebonded foam, 236 compression test, 88, 94, 105 compressive impact test, 108, 141

532 Index computational fluid dynamics, 184, 187, 190 contact stiffness, 509 core apparent shear modulus, 98 corner impacts, 302 crack growth, 355, 361, 366 crack growth direction, 365 crash mat, 236 creep, 91, 107, 298, 449 in closed-cell foams, 467 EPP, 475 FEA of Kelvin open-cell model, 463 gas diffusion contribution, 469 model for closed-cell foams, 470 model for high strain creep, 461 model for open-cell foam, 460 compliance, 450, 453, 457, 459, 461, 463 test, 110 crosslinked PE foam, 55 crosslinking, 49, 52 crushable foam model, 128, 130, 143 crystallinity, 41, 62, 309, 453 cushion curve, 108, 288, 289 design, 287 relation to stress strain curve, 291 test, 285 cyclic compression, 29, 224, 466 viscoelastic model for open-cell foam, 466 Darcy law, 481 density variation, 75, 79, 105, 106, 214, 359 gradient, 105, 357 diffusivity, 257, 259, 469, 472 gas, 12, 50, 60 thermal, 53 water, 484, 485, 497 drop height packaging, 283, 284 dynamic mechanical thermal analysis, 92 edge, 5 breadth, 6, 30 tensile response, 28 touching, 171 width, 5, 30, 159 effective impact energy, 509 effective mass, 509

elastic collapse stress, 155 Elastica, 160 elliptical integral, 161 energy absorption diagram, 292 EPP, 70, 72, 80, 98, 132, 133, 134, 135 EPS mass increase in water, 490 strength change in water, 494 EPS moulding design, 293 equilibrium density of expanding foam, 55 ESI, see ethylene styrene interpolymer ethylene styrene interpolymer, 510, 516, 517 Euler beam theory, 100 Euler-Bernoulli beam, 165, 461 Euler buckling, 155 Euler polyhedron formula, 8 EVA foam compression, 317 FEA, 320 impact durability, 322 microstructure, 311 processing, 311 replacements, 312 extrusion, 47 face, 55 curvature, 256 extension, 55, 56, 57 fracture, 12, 23, 47, 50 hole, 35, 182, 188 opening, 35, 178 orientation, 41 stress, 56 fatigue, 32, 322 FEA, 98 dynamic, 116, 140 foam model, 163, 169, 272 material model, 88, 117, 127, 238 packages, 116 static, 116 validation, 125 viscoelastic model, 241 filters, 200 foam deformation mechanisms, 148, 153, 252–3, 268, 275 dry foam, 3–5, 8 foam elastic anisotropy, 159 foam permeability, 179, 186, 187, 189, 257

Index foamability factor, 52 foot strike, 249, 314, 509 Fourier number, 260, 444, 484 fracture mechanics, 361 fracture toughness, 86, 110, 336, 358, 360, 362, 365, 368, 369 data for thermoplastic foams, 364 EPS, 367 and EPS fracture mechanisms, 369 model for open-cell foams, 363 fragility factor, 283, 284

gas diffusion constant, 258 gas volumetric strain, 256, 262, 475 Gent-Rusch model, 184, 186 Gibson and Ashby model, 153, 268 Gibson Ashby model for closed-cell foams, 268 model for open-cell foams, 153

hard segment, 24, 25, 28, 29 hardening, 98, 127, 129, 132, 136, 143, 156, 169, 450 head impact, 236, 243, 331, 526 heat transfer, 259 heat transmission in seating, 211 heelpad structure, 319 high resilience foam, 24, 30, 223, 522 hip effective mass, 413 hip bone structure, 405 hip fractures, 404 factors, 407 fall energy, 409 hip protector foam selection, 420 load shunting, 418 modelling of test rigs, 414 ranking, 416 shell stiffness, 421 test rigs, 410 types, 405 hydrolysis, 200, 229 hydrophilic foam, 488 hydrostatic compression test, 132 hyperelastic material, 89, 117, 125, 239 hysteresis, 28, 90, 122, 193, 198, 216, 224, 243, 312, 317, 324, 467

533

IFD, see Indentation Force Deflection in-package test, 301 indentation plane strain with a cube, 352 Indentation Force Deflection, 86, 217 indentation of EPS with large hemisphere, 355 with small hemisphere, 360 indentation stress ratio, 126, 219, 360 inertial flow coefficient, 179, 199 inertial flow resistivity, 179 irregular cell model, 13, 14, 167, 168, 274 isocyanate, 229, 428 isothermal, 255, 259, 262, 276, 291, 318 Kelvin foam, 6, 12, 13, 30, 80, 125 [001] structural cell, 266 air flow, 187 buckling, 162 closed-cell bulk modulus, 264 closed-cell elastic plastic model, 270 closed-cell inhomogeneous strain, 272 closed-cell Young’s modulus, 262, 276 compressive response, 160 diffusivity, 259 edge deformation, 166 edge moments of area, 152 model for foam expansion, 55 open-cell anisotropic, 466 anisotropy, 159 open-cell compression, 160–3 open-cell (wet compression), 163–7 open-cell creep, 461, 463 open-cell Young’s modulus, 156–8 relative density, 13 strain localisation, 171 wrinkled faces, 267 lifejacket, 499 linear acceleration, 284, 331, 332, 396, 415, 522 loading point yield, 102 long-range buckling, 162 martial arts mat, 236, 246 mastercurve, 243, 288 mattress, 213, 218, 223, 228 Maxwell model, 451

534 Index melt rheology, 48 molecular weight distribution, 82 Mooney-Rivlin equation, 119, 120 motorcycle helmet comfort foam deformation, 388 components, 374 deformation mechanisms, 384 effect of impact velocity, 391 EPS liner deformation, 386 FEA of direct impacts, 393 FEA validation, 397 liner, 378 lumped-mass model, 383 optimisation, 398 second impacts, 392 shell deformation, 386 shell, 376 test standards, 379, 381 moulded PU foam, 24 multi-axial stress test, 134 multiple impacts, 299 neutral surface, 86, 100, 101, 107, 431, 462 normalized stress, 155 normalized Young’s modulus, 154 oriented PP foam, 61 overlapping sphere model, 190 packaging, 281 pentane, 50, 59, 70, 71, 72, 441, 442 periodic boundary conditions, 157, 163, 168, 464 plane-strain indentation, 125, 127, 352, 353 plastic hinge, 253, 269, 271 Plateau border, 5, 80, 152, 159, 162, 484 Poisson’s ratio, 88, 120, 239 Poisson’s ratio test, 112 Poisson’s equation, 152 polar second moment of area, 151 polyethylene, 41 polyethylene foam, 62, 247, 272, 282, 303, 414 polypropylene, 46, 49, 519 polyurethane chemistry, 20 porosity, 199 pressure sore, 207, 208, 227 printer cartridge, 485 PVC foam, 428

rebonded foam, see remoulded foam relative density, 3 PE foam expansion, 57 relation to edge radius, 81 wet Kelvin foam, 13 wet Kelvin model, 6 relaxation time, 224, 241, 452, 466 remoulded PU foam, 239, 244, 246 FEA, 522 repeat impact, 323 representative unit cell, 153, 155, 163, 164, 187, 262 reticulated foam, 23, 486 rotational acceleration, 331 rubber shear modulus, 27, 119 rugby goal padding, 520 FEA, 522 foam, 522 head impacts, 523 rugby headguard, 526 sandwich beam, 432 sandwich panel bending stiffness, 431 boat building, 428 core crushing, 437 core shear failure, 437 local deformation, 435 torsional stiffness, 433 types, 426 seating pressure distribution, 209, 224 second moment of area, 102, 151, 431, 463 shear and compression test, 123 shear modulus, 88 of PU foam, 22 shell buckle, 421, 513 shin guard, 504 effectiveness, 507 FEA, 513 foam, 506 load spreading, 511 material selection, 505 standard, 508 stud impact, 510 test rig, 508 shoe ASTM test, 321 components, 308 features, 308

Index foam selection, 309 heel stability, 315 midsole, 308 midsole flexure, 315 pressure distribution midsole, 313 shrinkage post-extrusion, 59, 74 silicone surfactant, 23, 35 simple shear test, 86, 95, 110 single cell test, 150 slabstock polyurethane foam, 30, 169, 212 slabstock process for PU foam, 21 slow recovery PU foam, 33, 182, 455 soap foam shear modulus, 22 sound absorption, 198 sponge, 480, 485 steam, 71, 72, 73, 74, 75 stiffness, 87, 316 bending, 87 compressive, 87 torsional, 87 strain energy function, 117, 118, 119 strain localisation, 171 strain recovery, 300 stress distribution bending, 90 stress intensity factor, 361 Structural Insulation Panels, 426 Surface Evolver, 6, 159, 163, 192 surface tension, 20, 22, 54, 480, 481, 484 syntactic foam, 497, 498

tensile viscosity, 48 thermal conductivity, 441 effect of absorbed water, 494 thermal insulation panel, 440 thigh, 210 torsional compliance, 152 torsional stiffness, 151, 427, 433 tortuosity, 184, 199 transmissibility, 215, 216 truncated pyramid, 293, 296

tan δ, 52 tapered rib, 293, 298 design rules, 297 tensile stiffness, 433 tensile test, 99, 105, 131

X-ray diffraction, 26, 62 X-ray micro-tomography, 9, 169, 253

vertex, 6, 8, 80, 150, 157, 442 vertex reaction, 15 viscoelastic phenomena, 91 viscoelasticity, 449 non-linear, 465 Voigt model, 91 von Mises equivalent stress, 127 Voronoi model, 14–16, 167–8 water kinetics of uptake, 482 mass change, 482 permeation theory, 484 water absorption, 229, 494 types, 480 Weaire Phelan foam, 4, 13, 192 wet foam, 2–4, 6 wheelchair seat, 25, 212, 224 wiper foam, 488 wound care, 487

yield stress, 468 yield surface, 127, 129, 134, 137, 144

535

Polymer Foams Handbook: Engineering and Biomechanics Applications and Design Guide

Handbook of Polymer Foams

Handbook of Polymer Foams

Metal foams.. a design guide

Biomechanics. Principles and applications

Biomechanics: Principles and Applications

Metal Foams: A Design Guide

Biomechanics: Principles, Trends and Applications (Biomechanics: Theory and Applications)

Metal Foams: A Design Guide

Biomechanics Principles and Applications

Biomechanics: Principles and Applications

Biomechanics Principles and Applications

Biomechanics: Principles and Applications

Foams : theory, measurements, and applications

Polymer Materials Science And Engineering Handbook

Polymer Physics And Engineering

Biomechanics: Principles, Trends and Applications

Polymer Research and Applications

Emulsions, Foams, and Suspensions: Fundamentals and Applications

Emulsions, Foams and Suspensions - Fundamentals and Applications

Biomechanics: Principles, Trends and Applications

Handbook of Polymer Reaction Engineering

Handbook of Plastic Foams.. Types, Properties, Manufacture and Applications

Engineering Design Reliability Handbook

Standard Handbook of Biomedical Engineering and Design

Engineering Design Reliability Handbook

Engineering Biosensors: Kinetics and Design Applications

Cryptography Engineering: Design Principles and Practical Applications

Handbook of Plastic Foams

Foams: Theory: Measurements: Applications

Polymer Electrolytes: Fundamentals and Applications

Polymer Foams Handbook: Engineering and Biomechanics Applications and Design Guide

Handbook of Polymer Foams

Handbook of Polymer Foams

Metal foams.. a design guide

Biomechanics. Principles and applications

Biomechanics: Principles and Applications

Metal Foams: A Design Guide

Biomechanics: Principles, Trends and Applications (Biomechanics: Theory and Applications)

Metal Foams: A Design Guide

Biomechanics Principles and Applications

Biomechanics: Principles and Applications

Biomechanics Principles and Applications

Biomechanics: Principles and Applications

Foams : theory, measurements, and applications

Polymer Materials Science And Engineering Handbook

Polymer Physics And Engineering

Biomechanics: Principles, Trends and Applications

Polymer Research and Applications

Emulsions, Foams, and Suspensions: Fundamentals and Applications

Emulsions, Foams and Suspensions - Fundamentals and Applications

Biomechanics: Principles, Trends and Applications

Handbook of Polymer Reaction Engineering

Handbook of Plastic Foams.. Types, Properties, Manufacture and Applications

Engineering Design Reliability Handbook

Standard Handbook of Biomedical Engineering and Design

Engineering Design Reliability Handbook

Engineering Biosensors: Kinetics and Design Applications

Cryptography Engineering: Design Principles and Practical Applications

Handbook of Plastic Foams

Foams: Theory: Measurements: Applications

Polymer Electrolytes: Fundamentals and Applications

Recommend Documents