Metal Foams: A Design Guide

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64 Metal Foams: A Design Guide 6.2 Moments of sections A beam of uniform section, loaded in simple tension by a force, F, carries a stress D F/A where A is the area of the section (see Figure 6.1). Its response is calculated from the appropriate constitutive equation. Here the important characteristic of the section is its area, A. For other modes of loading, higher moments of the area are involved. Those for various common sections are given below and are defined as follows. The second moment of area I measures the resistance of the section to bending about a horizontal axis (shown as a broken line). It is � I D y 2 b⊲y⊳ dy section where y is measured vertically and b⊲y⊳ is the width of the section at y. The moment K measures the resistance of the section to twisting. It is equal to the polar moment of area J for circular sections, where � J D 2 r 3 dr section where r is measured radially from the centre of the circular section. For noncircular sections K is less than J. The section modulus Z D I/ym (where ym is the normal distance from the neutral axis of bending to the outer surface of the beam) determines the surface stress generated by a given bending moment, M: D Mym I D M Z Finally, the moment H, definedby � H D yb⊲y⊳ dy section measures the resistance of the beam to fully plastic bending. The fully plastic moment for a beam in bending is Mp D H y Thin or slender shapes may buckle locally before they yield or fracture. It is this which sets a practical limit to the thinness of tube walls and webs (see Section 6.5).
h SECTION h o h i d d o a b ;;; QQQ ¢¢¢ ;;; QQQ ¢¢¢ ;; ; QQ Q ¢¢ ¢ ; ; Q Q ¢ ¢ ;; QQ ¢¢ ;; ;;; QQ QQQ ¢¢ ¢¢¢ ;;; ; QQQ Q ¢¢¢ ¢ a d i a ; Q¢ bh b (h o − h i ) √ 3 4 a2 πd2 4 π 4 (d2 − d2 o i ) Figure 6.1 Moments of sections A (m2) I (m4) K (m4) H (m3) (m3) I ym bh 3 12 b 12 (h3 o − h3 i ) a4 32 √ 3 π 64 d4 π 64 (d4 − d4 o i ) 16 b hb 3 h 3(1−0.58 ) a 4 √ 3 80 π 32 d4 π 32 (d4 − d4 o i ) bh 2 6 b (h 12ho 3 o − h 3 i ) a 3 32 π 32 d3 π (d 32do 4 − d4 o i ) bh 2 4 b 4 (h2 o − h 2 i ) 1 6 d3 1 (d 6 3 3 − d o i ) Design formulae for simple structures 65
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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
Page 25 and 26: a) Preform Polymer ligaments d) Rem
Page 27 and 28: a) Vapor deposition of Nickel b) Bu
Page 29 and 30: Process Steps a) Powder / Can prepa
Page 31 and 32: Making metal foams 19 flight in a t
Page 33 and 34: a) Metal - Hydrogen binary phase di
Page 35 and 36: Entrapped gas expansion Making meta
Page 37 and 38: (a) (b) (c) 100µm Characterization
Page 39 and 40: E/E bulk σ peak /σ bulk 1.2 1.0 0
Page 41 and 42: Stress (MPa) Characterization metho
Page 43 and 44: Characterization methods 31 foam sp
Page 45 and 46: Characterization methods 33 ž Prop
Page 47 and 48: F/2 F/2 F/2 F/2 τ Core Characteriz
Page 49 and 50: Characterization methods 37 40 40 4
Page 51 and 52: Characterization methods 39 Gioux,
Page 53 and 54: (a) (b) (c) Properties of metal foa
Page 55 and 56: Table 4.1 (a) Ranges a for mechanic
Page 57 and 58: Stress, σ Schematic Young's modulu
Page 59 and 60: Properties of metal foams 47 foams.
Page 61 and 62: Properties of metal foams 49 show t
Page 63 and 64: Modulus 0.5 /Density (GPa 0.5 /(Mg/
Page 65 and 66: elations take the form P Ł � Ł
Page 67 and 68: Chapter 5 Design analysis for mater
Page 69 and 70: Table 5.1 Design requirements Desig
Page 71 and 72: Design analysis for material select
Page 73 and 74: 5.4 Where might metal foams excel?
Page 75: Table 6.1 Constitutive equations fo
Page 79 and 80: 6.3 Elastic deflection of beams and
Page 81 and 82: 6.4 Failure of beams and panels (a)
Page 83 and 84: ;; ;;; ;; M ;; ;;; ;; ;;; ;; ;; ;;
Page 85 and 86: Torsion of shafts T T,q T T,q Yield
Page 87 and 88: R a 2 Flow field R F R 2a 2 F F s c
Page 89 and 90: ;; ; QQ Q ¢¢ ¢ ;; ;; QQ QQ ¢¢
Page 91 and 92: ; ;; Creep p e p e • Area A u F 2
Page 93 and 94: and its deviatoric (i.e. shear) com
Page 95 and 96: Normalized effective stress 1.4 1.2
Page 97 and 98: A constitutive model for metal foam
Page 99 and 100: A constitutive model for metal foam
Page 101 and 102: Stress amplitude, ∆σ + Stress,
Page 103 and 104: Axial strain (%) 5 4 3 2 1 0 0 σma
Page 105 and 106: Nominal compressive strain Nominal
Page 107 and 108: σ max σ pl τ max τ pl σ max σ
Page 109 and 110: Design for fatigue with metal foams
Page 111 and 112: the net section stress criterion re
Page 113 and 114: E σ pl (m) 10 1 0.1 0.01 0.1 Relat
Page 115 and 116: Chapter 9 Design for creep with met
Page 117 and 118: Design for creep with metal foams 1
Page 119 and 120: instead to: � Pε 3 D Pε0 2 s Ł
Page 121 and 122: Time to failure (hours) 10 2 10 1 1
Page 123 and 124: Design for creep with metal foams 1
Page 125 and 126: 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,
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where D � 2⊲1 Eft 2 f ⊳GcR Sa
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Weight index ψ = W/2πR 2 ρ s W c
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Sandwich structures 145 plastic yie
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Sandwich structures 147 within the
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Sandwich structures 149 Shuaeib, F.
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Energy management: packaging and bl
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Energy/Unit cost (kJ/£) 10 1 0.1 0
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ys crushes axially at the load Ener
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peak pressure MPa Impulse/(mass of
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Chapter 12 Sound absorption and vib
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Sound absorption and vibration supp
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0 Sound absorption and vibration su
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Chapter 13 Thermal management and h
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Thermal management and heat transfe
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Thermal management and heat transfe
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~ P ~ Re 2.6 1000 800 600 400 200 0
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Chapter 14 Electrical properties of
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Electrical properties of metal foam
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Electrical properties of metal foam
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15.3 Joining of metal foams Cutting
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Pull-out load, F p (N) 10 5 10 4 10
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Cyclic loading of fasteners Cutting
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Time, material, energy, capital, in
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Cost estimation and viability 203 A
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$/Unit Melting Schematic of the liq
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Performance metric, P2 B Non-domina
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Cost estimation and viability 209 t
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Cost estimation and viability 211 i
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P 2 = 1/Loss coefficient (−) 1000
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Cost estimation and viability 215 C
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Chapter 17 Case studies Metal foams
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Case studies 219 Figure 17.2 A pres
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Case studies 221 Figure 17.4 Highwa
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Case studies 223 Figure 17.6 DUOCEL
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Case studies 225 density and high t
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Case studies 227 required for compa
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Maximum power density at electronic
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q *(MW/m 2) 15 10 Power density fix
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Case studies 233 which optimized sk
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e-mail: cymat@ican.net Web: www.cym
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Tel: C0043 7722 801-2125 Fax: C0043
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Chapter 19 Web sites An increasing
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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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