Page 1 of 66
`
`BOREALIS EXHIBIT 1039
`
`Page 1 of 66
`
`BOREALIS EXHIBIT 1039
`
`

`
`Cellular solids
`
`Structure and properties
`
`Second edition
`
`Lorna J. gibson
`Department of Materials Science and Engineering,
`Massachusetts Institute of Technology,
`Cambridge, MA 02139, USA
`
`Michael F. Ashby
`Cambridge University Engineering Department,
`Cambridge, UK
`
`
`
`Page 2 of 66
`
`Page 2 of 66
`
`

`
`PUBLISHED 131' THE mass SYNDlCA'I‘E OFTHI: UNIVERSITY or CAMBRIDGE
`The Pitt Building, Trumpington Street, Cambridge, CB2 nu» United Kingdom
`
`(‘,AMI3RlDGl;' UNIVERSITY PRESS
`The Edinburgh Building, Cambridge CB2 2RL, United Kingdom
`40 West 20th Street, New York, NY 10011-421 1, USA
`10 Stamford Road, Oakleigli, Melbourne 3166, Australia
`
`First edition (I) Lorna J . Gibson and Michael F. Ashby, 1988
`Second edition (0 Lorna J. Gibson and Michael F. Ashby, 1997
`
`This book is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without
`the written permission of Cambridge University Press
`
`First published by Pergamon Press Ltd., 1988
`Second edition published by Cambridge University Press, 1997
`
`Printed in the United Kingdom at the University Press, Cainbridge
`
`Typset in 10 '/4 / 13 ‘/zpt Times
`
`A catalogue rrer.nra'f0r this book is uvni/ablefram Hie Brilis/7 Librar_y
`
`Iiibrary of Congress" Ca ff!/aglll/lg in Publicatizm data
`Gibson, Lorna .l.
`Cellular solids : structure & properties /’ Lorna J. Gibson,
`Michael F. Ashby, ~ 2nd ed.
`p.
`cm. — (Cambridge solid state science series)
`Includes bibliographical references and index.
`ISBN 0-521-49560-1(hc)
`1. Foamed materials.
`11. Title.
`III. Series.
`T/\4I8.9.F(iG53
`1997
`62o.1'1—dc2o
`
`2. Porous materials.
`
`I. Asliby, M. F,
`
`96-31571
`cw
`
`ISBN o52i 49560 1 hardback
`
`
`
`Page 3 of 66
`
`Page 3 of 66
`
`

`
`Cambridge Solid State Science Series
`
`EDITORS
`
`Professor D. R. Clarke
`
`Department of Materials,
`University of California, Santa Barbara
`
`Professor S. Suresh
`
`Department of Materials Science and Engineering,
`Massachusetts Institute of Technology
`
`Professor 1. M. Ward FRS
`
`IRC in Polymer Science and Technology,
`University of Leeds
`
`l
`
`l N
`
`
`
`Page 4 of 66
`
`Page 4 of 66
`
`

`
`Material properties
`
`plastics. ceramics, glasses,
`F-‘omits can be made out of almost anything: metals.
`Ll on two separate sets of para-
`3' es. Their properties depen
`trie structure of the foam ~
`and even UOmDOHtl
`meters. There are those which describe the genme
`er is distributed between
`the size and shape oi" the cells, the way in which mutt
`thecell edgesand faces. and the relative density or porosity‘, these are described
`which describe the intrinsic properties
`in Clmptci' 2. And there are the parameters
`' lls are made‘, those we describe here.
`book are the den-
`Theeell w-.illprnpertlesw '
`'
`'
`‘
`nonlyin Lin‘:
`ol‘ the intiterial of w
`the Yming‘s modulus,
`I-.‘,.. the plastic: yield strength. rrys. the fracture
`'t‘l‘:ll expansion eoelficieiit, as,
`filly, pa.
`the thermal conductivity. /\.. the then
`at proI,3e1'ty of
`Thmttgltotit. the subscript ‘s’ itidlutttes
`strength, rm,
`"“" 1‘el'ers to a property of the
`and the specillc heat t;'.'.,5.
`tl while a super-.-n;1'iptetl
`‘the learn nmtlulus tlivicled by that of the cell
`the solid cell wall inateri:
`Itieztlis
`foam itself. Thus, 1:" /'J‘.'_.‘
`ative modulus’.
`wall material’; this is also rel'crrec.l to as ‘the rel
`materials, which
`ropertics til‘ solid
`It is helpful to start with an overview olthe 1.1
`*ang,es. A perspective on these is
`have values which lie in certain eha1'aete1‘istie I
`materialpi'upirr.=_r zrhtirts tAslil:y. 1992). ofwhich Figs. 3.1 and 3.2 are
`The first shows Ynung’s inetltilus, 13,,
`' against the density, ps-
`given by
`top right: they have high mciduli and high density. Fine cera-
`examples.
`licon carbide are stiller still, but. on a-verttge. a little less
`Metals lie near the
`metal. (llrisses he
`less dense than most
`mics like alumina or si
`sed on silica are
`envelope. Polymers
`itinnjttst to the left ml‘ the ‘metals’
`dense than the avemge
`duinina-.1tly mzide of light atoms, have the
`ceramics; they occupy a pens
`and elastomers, because they are pre
`
`52
`
`Page 5 of 66
`
`Page 5 of 66
`
`

`
`Introduction and synopsis
`
`53
`
`lowest densities of all solids and are much less stiff than the other classes because
`of the soft van der Waals bonding between the molecules. Polymer—matrix coin-
`posites, not surprisingly, lie somewhere between polymers and the glasses or car-
`bons with which they are reinforced. A general envelope for ‘polymer foams‘ is
`also shown for comparison (though here, of course, we show E‘, not ES). They
`lie at the extreme lower left.
`The second chart — Fig. 3.2 — shows the thermal expansion coelficient, as,
`plotted against the thermal conductivity, A. It reveals that polymers, and in parti-
`cular, elastomers, expand a great deal, and conduct heat very poorly - only
`foams (for which of and X‘ are plotted) have more extreme values ofthese proper-
`ties. Metals and certain ceramics conduct heat well and have relatively low values
`of as. Glasses and composites lie in between. Note the considerable range dis-
`played by solid properties » typically a factor of between 102 and I03 separates
`
`ID001-
`
`l. MODULU5‘DENSlTY
`VOUNGS MODULUS E
`(G=3E/5; K=E.l
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`DENSlTY, P (Mg/m3)
`
`Figure 3.1 A chart showing Young's modulus. E5, and density, ps, for materials.
`Each material class occupies a characteristic field on the chart.
`
`Page 6 of 66
`
`Page 6 of 66
`
`

`
`54
`
`Material properties
`the lowestfrom thelrighesl. But. even tlris is smallwhencompared withthe range
`which can be achievetl by l‘o;\111il1g_. which, for properties such as modulus and
`strength extendthe range by a l‘urther I‘-actor ofup to 104.
`The groupingsrevealed by these charts arethereasonthatweelecttodescribe
`material properties under three broad headings: polymers and elastomers,
`metals, and ceramics and glcrrses; this is done in the nextthree sections. Ineach,
`the constitutive equations used in analysing foam properties are developed. But
`totackle any practicalproblem,likethatofselecting afoamtopackage adelicate
`instrument or to serve as the core of a sandwich panel ~ one also needs data for
`the cell wall properties. Selected data for mechanical and thermal properties for
`solids are presented in Tables 3.1, 3.4 and 3.5; data for electrical and acoustic
`properties are given, whenrelevant, in the text ofthis chapter and in Chapter 7.
`Often (p:.irticularly for polymers) precise data for cell-wall properties may not
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`Page 7 of 66
`
`Page 7 of 66
`
`

`
`Polymers and elastomers
`
`55
`
`be available; then one has to estimate them. Methods for doing this are given in
`each section.
`
`There is an underlying difficulty in attempting to present a balanced overview
`ofmaterial properties in a single chapter — each ofits sections condenses inform a-
`tion which, adequately treated, requires a book to itself. As with pocket histories
`of Europe, or quick guides to World architecture, superficiality is a problem. It is
`minimized by focusing on specific goals. Ours are these: to introduce the cell—wall
`properties which are essential for the later development of models of cellular
`solids; to present typical data for these properties; and to provide sufficient physi-
`cal understanding that sensible estimates can be made for them when no data are
`available. But the reader who needs a comprehensive understanding of his or her
`material will need more than this: additional information will be found in the
`
`books listed in the reference section of this chapter under the special headin g ‘Ge n—
`eral references for compilations of material properties’.
`
`3.2
`
`Polymers and elastomers
`
`Engineering polymers have structures which, at first sight, seem impossibly com-
`plicated. Some are amorphous; others partly crystalline. In some the molecules
`
`are linear and tangled like lengths of string; in others, they are cross-linked like
`a fishnet. Some have a narrow spread of molecular weights; others a broad one.
`We will, as far as possible, avoid this complexity, calling only on those aspects
`of structure which directly relate to the mechanical and thermal properties. Prop—
`erties for typical polymers are given in Table 3. I. Polymer names and their abbre—
`viations are listed in Table 3.2.
`
`(a) Structure and density of polymers
`
`Roughly speaking, three sorts of polymer are used to make foams: thermoplas-
`tics, thermosets and clastomers. Linear t/iermoplasflcs like polyethylene (PE),
`polystyrene (PS) and ‘L)C|l_Vl‘13L'2lll)-'ll’IlCl2l‘laCI‘ylate (PMMA) have long, snake-like
`molecules. If the molecules have awkward side groups (like PMMA) which pre-
`vent neat packing, they solidify in an amorphous structure; if (like PE) they do
`not, they may coil together like fan—folded computer paper to form chain-folded
`crystals, though with amorphous material in between. The density of such linear
`Polymers is always near 1 Mg/m3.l That for low-density (largely amorphous)
`
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`Page 8 of 66
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`Page 8 of 66
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`Page 9 of 66
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`Page 9 of 66
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`Page 10 of 66
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`Page 10 of 66
`
`
`

`
`58
`
`Material properties
`
`Table 3.2 Common polymers and their short names
`Thermoplastics
`PE
`Polyethylene
`LDPE
`Low—density polyethylene
`HDPE
`High-density polyethylene
`PS
`Polystyrene
`PP
`Polypropylene
`PMMA
`Polymcthylmcthacrylatc (acrylic)
`PVC
`Polyvinylehloride
`PC
`Polycarbonate
`PA
`Polyamide (nylon)
`PVDF
`Polyvinylidene fluoride
`PTFE
`Polytetrafluroethylene
`PET
`Polyethyleneterephthalate
`
`Thermosets
`
`Elastomers
`
`EP
`UP
`PUR
`PF
`NR
`BR
`lR
`CR
`BUTYL
`PIB
`
`Epoxy
`Unsaturated polyester
`Polyurethane
`Phenolic
`Natural rubber
`Polybutadiene
`Synthetic polyisoprene
`Polychloroprene
`Butyl rubber
`Polyisobutylene
`
`polyethylene, for instance, is 0.92Mg/mg; that for polystyrene is l.05 Mg/m3.
`tly better packing; for polyethylene it increases the den-
`vier atoms
`(such as cellulose
`Crystallization gives sligh
`sity to 0.97 Mg/m3 (HDPE). Polymers with hea
`fluorine) are a little denser -
`which contains oxygen, or PTFE which contains
`their density can be as high
`2.2 Mg/ 1113.
`/inked Ihermosem. In these, strong
`The second class of polymers are the cross-
`-dimensional network
`C bonds link the molecules to form a three
`ther side groups where necessary to satisfy valence require-
`primary C
`f thermosets (epoxies, for instance) is usually larger than
`with hydrogen or 0
`tly because most contain heavy oxygen or nitrogen
`ments. The density 0
`he cross-linking itself increases the density by pulling
`that of thermoplastics. par
`The result is that most thermosets have densi-
`atoms, and partly because t
`the chains a little closer together.
`ties between 1.2 and l .5 Mg/mi‘.
`The final class is that of eIa.rtomcrs, or rubbers. Structurally, they can be
`thought of as linear—chain polymers with a few widely spaced cross—l
`
`inks attach-
`
`,
`
`
`Page 11 of 66
`
`Page 11 of 66
`
`

`
`
`
`Polymers and elastomers
`
`59
`
`mg E‘-d.(3l‘l molecule to its neiglibours. The tlillereiiee lies i.n the glass temperature, a
`material property which is its significant for aniorplinus solids as the melting
`point is for crystals. Unlike the first two classes nfpolymer, the glass temperature
`of an elastomer is well below room temperature. 'I"l1is means that at room ten»
`perature the secondary bonds have already melted, and the molecules can slide
`relative to each other with ease, Were it not for the cross—links, the material
`would be a viscous liquid, but the cross-links give it a degree of mechanical stabi-
`lity. Elastomers can undergo enormous deformations (500% is not uncommon);
`but on unloading the thermal motion of the molecules, coupled with the memory
`built into the arrangement of the cross-links, causes the material to return to its
`original shape. The restoring Force is a small one, so the moduli ol'e1astorners
`are less, by a factor of 1000 or more, than those ofthermoplastics or thermosets.
`The packing of the loosely tangled molecules is similar to that ofnon-crystalline
`linear, polymers, so the density of most elastomers is, like that of polyethylene,
`about l Mg/m3 (Table 3.1).
`When the cell-wall density of a polymeric foam is not known, it can be esti-
`mated, with an error which is unlikely to exceed L1 5%, by selecting the appropri-
`ate line in Table 3.3. Its basic features have already been explained. Amorphous
`polymers containing only carbon and hydrogen always have a density near
`lMg/m3. Heavier atoms (the commonest are oxygen, chlorine and fluorine)
`increase the density: teflon ~ (CF2),,» with two fluorines per carbon, is con-
`siderably denser than polyethylene, ~(CH2)~. As a rule of thumb the density
`is roughly (1 + 0.7 m) Mg/m3, where m is the number of heavy atoms (meaning
`0, N, F or Cl) per carbon in the polymer chain. Cross-linked polymers tend to
`be denser than simple linear polymers for two reasons given earlier: they almost
`always contain heavy atoms, and the cross-linking draws the atoms closer
`together. The polymers on which plant-life is based, containing a large fraction
`of cellulose, are denser still, largely because the molecules in crystalline cellulose
`
`Table 3.3 Estimates olr>polymer density
`
`Type of polymer
`
`Linear polymers not containing heavy atoms
`Simple elastomers (latex)
`Linear polymers containing in heavy atoms (F, Cl, 0) per carbon
`Simple Cross-linked polymers (epoxies, polyesters)
`Plant polymers based on cellulose and lignin
`
`Approximate
`density
`(Mg/m3)
`
`1.0
`l.0
`(1 + 0717:)
`1.4
`1.5
`
`Page 12 of 66
`
`Page 12 of 66
`
`

`
`60
`
`Material
`
`P
`
`P
`ro erties
`
`contain oxy gen,
`trees and smaller
`
`and pack efficientl
`plants is close to
`
`y; the average density of the cell wall of both
`l .5 Mg/ 1113 .
`
`ties of polymers
`(1)) Thermal proper
`they (‘.l1‘.1l‘lgC
`1 they melt. when
`:1re solid Ltt‘ll;l
`It is common exne1‘ienu:.
`e that stalltls
`mlerystnlline linenr polyrners do
`Antorpltotis '.-ind st:
`form l"1'ont ll VlSCl‘JEl'.'ttitlI.: solid
`ttiseontlnuously to ltqttltls.
`tl. they tratns
`above which they
`not have it sharp mumng pumr: l'l1h'l.t':[l
`gt: of te1.npe1'tLtu1‘t-:
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`to L1 viscous llqtlltl over tt \\’L‘ll-tl'L‘.lll'l.t:t.l retn
`ezm he moultled or foztiwcd. ('1-oss~li.nked he-lytnei's tlo no
`ramn-e is ntiseti too
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`'!'g,
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`tn.-i-i.\-iii'm1 tempt-rmiiru.
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`froth the solitl to ruhbe1' or viscous llquitl.
`l'1'on1 our point 0
`Below TL: the::l1t'lL‘.|1‘1|')l'l0Lt5componentol‘thepolymer is glassy. Whenitislott-.leLl_
`which ehttt‘ttete1"t'.Les the tr:1ns'1tton
`it tltstorts elastically and then yields or h".-tethres; it l"oan't 1Tl':'.L'lL‘. 0|‘ it would he
`Ll't*lgiLl‘. Above TR thepolymer isviseoelttstleo1'elt1stornet'ieandcansttstttln
`n mctwerahle, tlel'ornta.tions without true yieltl or l'i':teture: a foam
`nlletl ‘flexible’. Most eommereittl nolymersliawe Id gl;I.sstetn-
`lttrgc. otte
`as shown in Tnble 3. t. The glass t1'nnsit'ton lEl'l‘l—
`in-utle til" it would he as
`and with the degree ol‘ ei'oss-lhtklttg.
`tlte rttnge tl~ltlt1‘ C.
`pe1'2ttnre in
`tses with nmleettlur weight
`t of -,1olymt-.-r SC'fI’.t1'Lt'.I:5 :: yohtms-1;-.t teinpemture T,‘ from one t-ll it
`perature inure:
`hem [lows troin the hotter side to the eolde: one. ll‘ this
`When at slice
`§'.l1..\‘lf.-‘tftl'_|-'t'l‘l'tH(’ tsestabltshetl. Then the heat flux. q
`lower tetnpemtttre T3.
`to the slieet tl1iekm:.ss I nntt the temper:1ttn'e d'1t‘l'er-
`tn'r:tngc1net1t is lelit tong ertou
`(me:tsu1'ed in W I mg‘) is related
`Tfihy
`enee 7'1 —
`
`ettlle
`
`l
`
`I
`
`- T2)
`l'_’L_
`q:>‘s'
`1
`is applied sutltlenly to one
`. lint when l.’l€.Etl
`heat tlillhses ll'lW':tt't_lS at a
`S is the H'm‘nml L-unu‘m'.‘i'w‘I_t-'
`are !."r.‘.f'J.l'fl'fH'. the
`hermal
`the sheet. the conditions
`S. It is related to the t
`surface of
`the Ih:'rmnl' itijflii.-rft'i!_ti. u
`is detcrintuetl hy
`rate which
`conductivity by
`A.“
`4..
`ptCps
`
`as = ~
`
`(3.1)
`te ltcttt. I-lettt inerettses the loeul ampli-
`N is the smell‘
`in the polymer e.htt'tn.
`is the tlcnsity '.1l‘|Ll ( ‘F,
`hieh '.u'e strung together
`l‘ the ‘atom.-1 w
`relies on the f!Tt)|.Td.}:’,‘¢t~ettt
`tucle 01' vihi'ttti<_n'1 o
`e lt‘:i'.‘l.\'pUt‘1. or eiiergy.
`l1it:.l1 imrolves Ill
`l3t‘ttl.lt,)l1$‘. their etfielettey in earrying h
`He-in L‘t‘I1‘ltl'LlCll01'1, w
`lllng lnttiee vi
`toms are entipletl (and thus
`tion of pltonons — trzwe
`ratltotts ol‘tu'lj:tt:e1tt:t
`free-patlt - the dis-
`n how sttmtgly the vlh
`the pltonon Incan-
`tlepemls o
`lent bonds in :1 crys-
`)L'lLIl\lS cat" the mnteri'.=tl) and on
`ectetl. ("own
`no the 'l‘l'lt
`before it ls st-.:itte-rett or well
`tt-thee El phonon moves
`
`whcve {J5
`
`
`
`Page 13 of 66
`
`Page 13 of 66
`
`

`
`Polymers and elastomers
`
`61
`
`tnllogmpliieally ot‘deretl array give high moduli and strong coupling; and in very
`perfect crystals phnnons travel at long way hefore they are scattered [so diamond.
`for ll'lSltlI'lL'(.‘. is a very good tliermtil etmtluctor). Weak. van der Wuals bonds
`give lnw moduli and weal-L coupling. t't1'lE.l the disordered structure elia|‘neteI'istie
`of virtually all polymers. gives it mezurfree-patli which is only three or four
`atom-dianneters in length. That is why polymers have thermal conductivities
`which are lower than those ul".n1y other class of dense solid (Fig. 3.2).
`As it general rule. the thermal enntluctivities ofpolymers lie in the surprisingly
`narrow range 0.3 i (H W fn1.l{; it‘ no data are available. this value e-an he used.
`Polymers with some erys1'.1lliI1it_v (such as PE, PP, PTFE.) lie at the high end of
`the range; amorphous polymers like PMMA and the elastomers polyisoprene
`:1nd polybutadiene, til the lower end. The thermal conductivity changes with tem-
`perature hy less tlran WM. between t] and l0O°C, so the room temperature value
`is usually adequate.
`The .t'p<'r.€fiC heat CPS, is the energy reqLllred to raise the temperature ofthe unit
`mass by l''('. The value Lil" ("IE depentls‘. on the vibration and rotational motions
`excited in the solid, and this in turn is mainly dictated hy the chemical structure
`til‘ the polymer. The specific hen ts of all rigid polymers are large; they lie in the
`range l_2—1 .tll<.JHl<.gK: :1 value in the middle ol‘ this range is the best estimate it"
`no tltttn are awtilable. Elnstomers have still larger specific heats; the range, typi-
`cally, is 1.8-2.5 kJ/l<gK.
`As the material property chart of Fig. 3.2 showed, polymers as a class show
`higher coeficients ofthermal expansimi. :.u,,, than metals or eeratnies. The dilTer~
`ences between polymers are not well understood. Rigid polymers have values in
`the range StJ—l Sli x 10-6,’ K; the middle of this range is a usable estimate if no
`data are available. Elastomers can have much larger values: up to 500 x 10-6/K.
`
`(c) Elastic properties of polymers
`Polymers show three distinct regimes ofelustic behaviour: the glassy 1‘egi1i1e,tl1e
`glass transition and the fiibber regime (Fig. 3.31. Each has certain well-defined
`characteristicsillustrated. for a thermoplastic, at thernioset and an elrustnmer, in
`Figs. 3.4, 3.5 and 3.6. The axes of these figures are the modulus ifs, and the nor-
`malized temperature T/ Tg, where Tg is the glass temperature. Each figure is
`divided into fields enrresponding to the main regimes of behaviour. Contours
`show the value ofthe relaxn Lion modulus at various loading 1imes.l The modulus
`Varies widely: between absolute zero and the temperattwe at which the polymer
`decomposes or melts (typically,
`l50~300°C) the modulus of a thermoplastic
`
`llhe relaxation modulgs at a time I is the stress a at 1 divided by the strain 5 in a
`stress relaxation test.
`
`Page 14 of 66
`
`Page 14 of 66
`
`

`
` ‘
`
`Mechanical properties of foams: compression
`
`183
`
`generates a foI'ee which must also be overcolne. This, and lime-dependent prop-
`erties (including. creep and viscoelasticityl we leave to Chapter 6.
`Each of these mechanisms is analysed in the following sections, giving equa-
`tions which describe the mechanical characteristics of each class of foam.
`
`5.3
`
`Mechanical properties of foams: compression
`
`Most applications of foams cause them to be loaded in compression, the subject
`of this section. Their behaviour in tension is discussed in Section 5.4.
`
`(a) Linear elasticity
`The linear elastic behaviour of a foam is characterized by a set of moduli. Two
`moduli are needed to describe an isotropic foam; usually they are chosen from
`the Young’s modulus, E‘, the shear modulus, G*, the bulk modulus, K*, and the
`Poisson’s ratio, 12*. More are required for one that is not isotropic: five if the
`structure is axisymmetric; nine if it is orthotropic. Here we give expressions for
`E”, G*, K*, and 12*, for an isotropic foam in terms of the cell—wall modulus, Es,
`and the relative density, p‘ /ps of the foam itself (anisotropy is discussed in Chap-
`ter 6). The literature contains numerous attempts to do this (Gent and Thomas,
`1959, 1963; K0, I965; Chan and Nakamura, 1969; Patel and Finnie, I970; Leder-
`man, 1971; Menges and Knipschild, I975; Barma et al., I978; Gibson and
`Ashby, 1982; Christensen, 1986; and Warren and Kraynik, 1988). Much of it
`makes diflicult reading, partly because of a confusion about the mechanism of
`linear-elastic deformation. and partly because authors often select a particular
`cell shape (connected struts forming a rhombic dodecahedron, for example) and
`seek a |'nll analysis of its response to stress.
`The meclianistn of linear elasticity depends on whether the cells are open or
`closed. At low relative densities, open-cell foams deform primarily by cell-wall
`bending (Kn, I965; Patel and Finnie, I970; Mcugcs and Knipschild, I975; Gib-
`son and Ashby,
`I982; Warren and liiuynik,
`I9!-ll-1}. As the relative density
`increases tn‘ /p_, *.»'il.l) the eonti-ibtition of sirnplc: extension or eornpressiun ol‘
`the cell walls becomes more signilicnnt (Warren and I(I.':.-tynik. I988). Flttid llow
`through an open-cell foam usually only cont1'ib'utes to the elastic moduli if the
`lluid has it high viscosity or the strait]-I'ate is exeepljotially high; this ell’ecl
`is
`described in Chapter ti. III closed-cell foams the cell edges both bend and extend
`or coI'ItI':IcI. while the i11en'.hI'aIies which Form the cell faces stretch. iiicrettsing
`the eon'1ribI.ition of the axial cell-wall stiffness to the elastic moduli (Gem and
`Tlminas, I959, I9n3; Pt-l1'elz1]1ClFl.l1l‘llC, I970; Letterman, 1971: Green, I985; and
`Cl'Il‘lSl.C]ZlSC1'1. I986).
`ll" the memhrauies do not rtipture. the l'.'£.‘.'l’l‘.]‘3l'eSl:‘Il¢.)['I of the
`cell fluid which is trapped within the cells also increases their stifincss. Each of
`
`Page 15 of 66
`
`Page 15 of 66
`
`

`
`_
`
`— _
`
`:84
`
`The mechanics of foams: basic results
`
`the nieelialiisins wl1ie|~. contribute to the lin'e:11'-elastic response offoams is shown
`seliesmltleally in Fig. 5.5. We analyse the respuu.-ze 01' both open- and closed-cell
`Foams to load using simple mechanics and avoiding dillieull geometry by using
`scaling laws.
`
`Open cells
`At the simplest level an open—cell foam can be modelled as a cubic array of mem-
`l3t'.‘Tfi oi’ length I Lind sqtr.1re L'.F0.‘s'.S—tit3t:ll(‘.ll‘| of side 1 [Fig. 5.6), Adjoining cells are
`s1agge1'cd so that their members meet at their niidpoiiits. The cell-shapes in real
`[hams are. ofcourse, more complex than that of Fig. 5.6. But ifthey defmm and
`I‘;-til by the same meelianisms. their properties can be understood using dimen-
`sional arguments which omit all ‘constants arising from the specific cell geometry.
`The relative density of the cell. ,r:',';1,._, and the second moment of area of a mem-
`ber, 1. are related to the dimensions r and 1 by
`:-1-ca
`
`and
`
`(5.2)
`I (X I4.
`Young’s modulus for the foam is calculated from the linear-elastic deflection
`oi‘ a beam oflength I loaded at its midpoint by a load F. Standard beam theory
`(e.g. Timoshenko and Goodicr, 1970) gives this dcilection, 5, as proportional to
`
`F, 5
`
`F_ 8
`
`Figure 5.5 The
`mechanisms of
`deformation in foams. (ai)
`open—cel1 foams ~ cell-wall
`bending + cell-wall axial
`deformation + fluid flow
`between cells. (b) closed-
`cell foams - cell-wall
`bending + edge contraction
`and membrane stretching
`+ enclosed gas pressure.
`
`( O )
`
`
`
`Page 16 of 66
`
`Page 16 of 66
`
`

`
`'
`
`‘
`
`'
`
`
`
`
`
`‘
`
`|
`i
`
`185
`
`Mechanical properties of foams: compression
`F13/E51, where E5 is the Young’s modulus for the material ofthe beam. When a
`uniaxial stress is applied to the foam so that each cell edge transmits a force F
`(Fig. 5.7) the edges bend, tmd the linear-t.:la:.st'i:: delleulioti of the structure as at
`whole is proportional to F!’ /E51. The force, F. is related to the remote compres-
`sive stress, a, by F VX all and the strain r is related to the dispiztcemenl. 5, by
`6 oc 6/ I. It follows immediately that Young's modulus for the foam is given by
`E,
`or
`CIESI
`
`Figure 5.6 A cubic model
`for an opcn—cell foam
`showing the edge length, I,
`
`and the edge thickness, 1.
`
`F
`
`
`
`i
`
`Figure 5.7 Cell edge
`bending during linear-
`elastic deformation.
`
`CELL EDGE
`BENDING
`
`Page 17 of 66
`
`Page 17 of 66
`
`

`
`186
`
`The mechanics of foams: basic results
`
`from which (using Eqns. (5.1) and (5.2))
`
`E,
`133
`
`1 : C, <-)
`
`/J.t<
`95
`
`2
`
`(open cells)
`
`(5.3)
`
`includes all of the geometric constants of proportionality. Any
`where Cy
`equiaxed cell shape leads to this result; the only dilference lies in the constant
`C1. Data, discussed later, show that C, m 1.
`Some ceramic foams are made by infiltrating a slurry of ceramic powder into
`an open-cell polymer foam and then firing the ceramic. The firing process yields
`a porous ceramic strut and burns ofi" the polymer, leaving a central void in the
`strut in place of the polymer. Measurement ol‘ the Young’s moduli of such alu-
`mina foams indicate that the data are well fitted by Eqn. (5.3) with C] : 0.3
`(Hagiwara and Green, 1987). Here, E, is taken to be the Young’s modulus of
`the fully dense solid making up the strut; the constant of proportionality, C1, is
`reduced by the porosity within the ceramic strut. It is interesting to note that a
`hollow tubular strut with no porosity in the solid resists bending more efficiently
`than a fully filled strut; then we would expect the constant C1 to be greater than
`one.
`
`This, it must be emphasized, is the modulus at small strains. As the elastic dis-
`tortion increases, the axial load on a cell edge (which we shall call P) increases,
`too. If P reaches the Euler load, Pm, for the edge, it buckles; that is a problem
`we analyse in Section 5.3b. But even before it buckles the axial load exerts an
`additional moment (which, till now, we have neglected) on the bent edge. In coin-
`prcssion this bcam—eolumn interaction lowers the modulus, E‘;
`in tension
`it increases it. So the part of the stress strain curve that we have called ‘linear-
`elastic’ is not truly linear. but is concave downwards; and Young’s modulus, if
`measured at finite strain, is smaller in compression than in tension.
`The shear modulus is calculated iii a similar way. lfa shear stress, 7, is applied
`to a foam, the cell members again respond by bending. Since the bending de-
`flection, 6, is proportional to F13/E51, and the overall stress, T, and strain, *y, are
`proportional to F//2 and 6/[, respectively, we find
`
`G, : I : CZESI
`7
`I“
`
`from which
`
`2
`
`(open cells)
`
`<9->
`s -
`pg
`125
`for the equiaxed shape. Data shown below suggest that C; m 3/8.
`Poisson’s ratio, 1/‘, is the negative ratio of the lateral to the axial strain. Since
`both are proportional to a bending deflection per cell length, their ratio is a con-
`stant, and for this reason Poisson’s ratio is solely a function of cell geometry
`and is independent ofdensity. For a material which is linear-elastic and isotropic
`
`(5.4)
`
`Page 18 of 66
`
`Page 18 of 66
`
`

`
`
`
`Mechanical properties of foams: compression
`
`I87
`
`E
`G =;
`2(l +7/)
`
`from which
`
`it 7 Cl
`V — 2C2
`
`_
`1— C3-
`
`I-
`
`(5-5)
`
`As with honeycombs, foams with a negative Poisson’s ratio can be made by
`inverting the cell walls to make a structure which is the equivalent, in three
`dimensions, ofthc inverted shape shown in Fig. 4.3(b). Here, the effect arises (as
`pointed out by Gibson et a1., 1982) from the change of cell geometry associated
`with the flexural deformation of the cell walls. The inverted cell shape (Fig. 5.8
`and Fig. 5.9(a)) can be produced from conventional foams in a variety of ways
`(Lakes. 1987; Friis, Lakes and Park, 1988; Lakes, 1991; and Choi and Lakes,
`1992). Thermoplastic foams can be triaxially compressed at higher temperatures
`and then cooled in the deformed state so that the ribs of each cell permanently
`
`Figure 5.8 A foam with
`v’ : -0.5. Poisson‘s ratio
`is negative because the
`structure has re-entrant
`walls. the three-
`dimensional equivalent of
`Fig. 4.3b (Lakes, 1987).
`
`
`
`(C)
`(b)
`(0)
`Figure 5.9 Microstructures giving negative Poisson‘s ratios. (a) Inverted ‘re-
`entrant’ cell shape. (b) Solid cylinders or spheres attached to each other by elastic
`strips. (C) Nodes. connected by tensile springs. constrained by hinged incxtensible
`rods.
`.
`
`
`Page 19 of 66
`
`Page 19 of 66
`
`

`
`188
`
`The mechanics of foams: basic results
`
`pettetrzttc inwards. Thermosetting silicone rubber l‘oarns can he tri-axially com-
`pressed during the curing pI't:t:ess. And conventional metal forums can he plasti-
`cally de|'ormed setutentially in each of three ortlmgonal directions. The re-
`enlr:=mt cells that result give i“nisso1t‘s ratios lying between D and ~tl.f~'.
`I_.al-tes.
`Negative Poisson‘s ratios c;-tn arise in other ways (A|mg1‘eu. 1985'.
`lt)l'i-7.
`tt}88. tggt. |fi)t).‘£t‘t,i'5, I993; Laltes ¢.'t‘ ul., I988. Evans, Julia): 1990: Evans er
`m'.. 1992: Evans and Catltlocl-t. "1989: (‘adtltiek and Evans. 19?-<9". WarretL 1.990:
`Wei. I992}. Two are shown in Flifli. 5.901) and (C). The [irst (Lakes. 1991) shows
`solid cylinders or spheres attaeltetl to each other by thin elastic strips or wires.
`When the strut:t1Ire is strt:tL:.l1ctl
`in the direction of the arrows. the ligaments
`unwrap from the cylinders or spheres. causing them to rotate; in doing so, the
`structure exptt