Metal Foams: A Design Guide

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192 Metal Foams: A Design Guide with that of the edges, and since only one third of these conduct in a body containing a fraction / s of conducting material, the relative conductivity is simply � � D s 1 ⊲14.5⊳ 3 s where s is the conductivity of the solid from which the foam was made. As the relative density increases, the nodes make an increasingly large contribution to the total volume of solid. If the node volume scales as t3 and that of the edges as t2ℓ, then the relative contribution of the nodes scales as t/ℓ, oras⊲ / s⊳1/2 . We therefor expect that the relative conductivity should scale such that s D 1 3 � D 1 � 3 s s � � � 1 C 2 � C 2 � 3 s s � 3/2 � 1/2 � ⊲14.6⊳ where the constant of proportionality 2 multiplying ⊲ / s⊳1/2 has been chosen to make / s D 1when / s D 1, as it obviously must. Real foams differ from the ideal of Figure 14.3 in many ways, of which the most important for conductivity is the distribution of solid between cell edges and nodes. Surface tension pulls material into the nodes during foaming, forming thicker ‘Plateau borders’, and thinning the cell edges. The dimensionality of the problem remains the same, meaning that the fraction of material in the edges still scales as / s and that in the nodes as ⊲ / s⊳3/2 , but the multiplying constants depend on precisely how the material is distributed between edges and nodes. We therefore generalize equation (14.6) to read � � � �3/2 D ˛ C ⊲1 ˛⊳ ⊲14.7⊳ s s s retaining the necessary feature that / s D 1when / s D 1. This means that the conductivity of a foam can be modeled if one data point is known, since this is enough to determine ˛. Equation (14.7) is plotted on Figure 14.2, for two values of ˛. The upper line corresponds to ˛ D 0.33 (the ‘ideal’ behavior of equation (14.6)), and describes the open-cell Duocell results well. The lower line corresponds to ˛ D 0.05, meaning that the edges make a less than ideal contribution to conductivity. It fits the data for Alulight well. Alulight is an closed-cell foam, yet its behavior is describe by a model developed for open cells. This is a common finding: the moduli and strengths of closed-cell foams also lie close to the predictions of models for those with
Electrical properties of metal foams 193 open cells, perhaps because the cell faces are so thin and fragile that they contribute little to the properties, leaving the edges and nodes to determine the response. References Babjak, J., Ettel, V.A., Passerin, V. (1990) US Patent 4,957,543. Gibson, L.J. and Ashby, M.F. (1997) Cellular Solids, Structure and Properties, 2nd edition, p. 296, Cambridge University Press, Cambridge. Kraynik, A.M., Neilsen, M.K., Reinelt, D.A. and Warren, W.E. (1997) In Sadoc, J.F. and River, N. (eds), Foam Micromechanics, Foams and Emulsion, p. 259, Proceedings NATO Advanced Study Institute, Cargese, Corsica, May. Lemlich, R. (1997) J. Colloid and Interface Science 64(1), 1078. Mepura (1995) Data Sheets, Metallpulvergesellschaft m.b.h. Randshofen A-5282, Braunau, Austria. Phelan, R., Weaire, D., Peters, E. and Verbist, G. (1996) J. Phys: Condens Matter 8, L475.
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Metal Foams: A Design Guide
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Copyright © 2000 by Butterworth-He
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vi Contents 4.3 Foam property chart
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viii Contents 17 Case studies 217 1
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List of contributors M.F. Ashby Cam
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Table of physical constants and con
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Chapter 1 Introduction Metal foams
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Activity Research Industrial take-u
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Application Comment Introduction Bi
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Cell size (cm) Making metal foams 7
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Making metal foams solidify. The th
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Making metal foams 11 2.4 Gas-relea
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a) Preform Polymer ligaments d) Rem
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a) Vapor deposition of Nickel b) Bu
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Process Steps a) Powder / Can prepa
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Making metal foams 19 flight in a t
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a) Metal - Hydrogen binary phase di
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Entrapped gas expansion Making meta
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(a) (b) (c) 100µm Characterization
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E/E bulk σ peak /σ bulk 1.2 1.0 0
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Stress (MPa) Characterization metho
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Characterization methods 31 foam sp
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Characterization methods 33 ž Prop
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F/2 F/2 F/2 F/2 τ Core Characteriz
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Characterization methods 37 40 40 4
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Characterization methods 39 Gioux,
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(a) (b) (c) Properties of metal foa
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Table 4.1 (a) Ranges a for mechanic
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Stress, σ Schematic Young's modulu
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Properties of metal foams 47 foams.
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Properties of metal foams 49 show t
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Modulus 0.5 /Density (GPa 0.5 /(Mg/
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elations take the form P Ł � Ł
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Chapter 5 Design analysis for mater
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Table 5.1 Design requirements Desig
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Design analysis for material select
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5.4 Where might metal foams excel?
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Table 6.1 Constitutive equations fo
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h SECTION h o h i d d o a b ;;; QQQ
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6.3 Elastic deflection of beams and
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6.4 Failure of beams and panels (a)
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;; ;;; ;; M ;; ;;; ;; ;;; ;; ;; ;;
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Torsion of shafts T T,q T T,q Yield
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R a 2 Flow field R F R 2a 2 F F s c
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;; ; QQ Q ¢¢ ¢ ;; ;; QQ QQ ¢¢
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; ;; Creep p e p e • Area A u F 2
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and its deviatoric (i.e. shear) com
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Normalized effective stress 1.4 1.2
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A constitutive model for metal foam
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A constitutive model for metal foam
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Stress amplitude, ∆σ + Stress,
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Axial strain (%) 5 4 3 2 1 0 0 σma
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Nominal compressive strain Nominal
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σ max σ pl τ max τ pl σ max σ
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Design for fatigue with metal foams
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the net section stress criterion re
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E σ pl (m) 10 1 0.1 0.01 0.1 Relat
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Chapter 9 Design for creep with met
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Design for creep with metal foams 1
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instead to: � Pε 3 D Pε0 2 s Ł
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Time to failure (hours) 10 2 10 1 1
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Design for creep with metal foams 1
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Chapter 10 Sandwich structures Sand
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Sandwich structures 115 The elastic
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Indentation Sandwich structures 117
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H H Core shear Core shear Collapse
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0.5 10−1 t c t c 10 −2 0.5 10
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Sandwich structures 123 the contact
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Sandwich structures 125 four sectio
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c t p p Figure 10.9 Sandwich panel
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Sandwich structures 129 with k ¾ D
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c/ Face sheet yielding 0.14 0.12 0.
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Weight index, ψ 10 −1 10 −2 10
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Sandwich structures 135 To construc
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Sandwich structures 137 The associa
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Sandwich structures 139 Note that,
Page 153 and 154: where D � 2⊲1 Eft 2 f ⊳GcR Sa
Page 155 and 156: Weight index ψ = W/2πR 2 ρ s W c
Page 157 and 158: Sandwich structures 145 plastic yie
Page 159 and 160: Sandwich structures 147 within the
Page 161 and 162: Sandwich structures 149 Shuaeib, F.
Page 163 and 164: Energy management: packaging and bl
Page 167 and 168: Energy/Unit cost (kJ/£) 10 1 0.1 0
Page 171 and 172: ys crushes axially at the load Ener
Page 179 and 180: peak pressure MPa Impulse/(mass of
Page 183 and 184: Chapter 12 Sound absorption and vib
Page 185 and 186: Sound absorption and vibration supp
Page 189 and 190: 0 Sound absorption and vibration su
Page 193 and 194: Chapter 13 Thermal management and h
Page 195 and 196: Thermal management and heat transfe
Page 197 and 198: Thermal management and heat transfe
Page 199 and 200: ~ P ~ Re 2.6 1000 800 600 400 200 0
Page 201 and 202: Chapter 14 Electrical properties of
Page 203: Electrical properties of metal foam
Page 207 and 208: 15.3 Joining of metal foams Cutting
Page 209 and 210: Pull-out load, F p (N) 10 5 10 4 10
Page 211 and 212: Cyclic loading of fasteners Cutting
Page 213 and 214: Time, material, energy, capital, in
Page 215 and 216: Cost estimation and viability 203 A
Page 217 and 218: $/Unit Melting Schematic of the liq
Page 219 and 220: Performance metric, P2 B Non-domina
Page 221 and 222: Cost estimation and viability 209 t
Page 223 and 224: Cost estimation and viability 211 i
Page 225 and 226: P 2 = 1/Loss coefficient (−) 1000
Page 227 and 228: Cost estimation and viability 215 C
Page 229 and 230: Chapter 17 Case studies Metal foams
Page 231 and 232: Case studies 219 Figure 17.2 A pres
Page 233 and 234: Case studies 221 Figure 17.4 Highwa
Page 235 and 236: Case studies 223 Figure 17.6 DUOCEL
Page 237 and 238: Case studies 225 density and high t
Page 239 and 240: Case studies 227 required for compa
Page 241 and 242: Maximum power density at electronic
Page 243 and 244: q *(MW/m 2) 15 10 Power density fix
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Page 247 and 248: e-mail: cymat@ican.net Web: www.cym
Page 249 and 250: Tel: C0043 7722 801-2125 Fax: C0043
Page 251 and 252: Chapter 19 Web sites An increasing
Page 253 and 254: http://www.seac.nl/english/recemat/
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(b) (continued) Appendix: Catalogue
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(f) Vibration-limited design Append
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Index (Company and trade names are
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Heat transfer 4, 182 Heat transfer
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Surface strain mapping 36 Syntactic
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Metal Foams: A Design Guide

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