Lecture 7, Foams, 3.054
Open-cell foams
• Stress-Strain curve: deformation and failure mechanisms
• Compression - 3 regimes - linear elastic - bending
- stress plateau - cell collapse by buckling
yielding
crushing
- densification
• Tension - no buckling
- yielding can occur
- brittle fracture
Linear elastic behavior
• Initial linear elasticity - bending of cell edges (small t/l)
• As t/l goes up, axial deformation becomes more significant
• Consider dimensional argument, which models mechanism of deformation and failure, but not cell
geometry
• Consider cubic cell, square cross-section members of area t2 , length l
1
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figures courtesy of Lorna Gibson and Cambridge University Press.
2
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
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Foams: Bending, Buckling
Figure removed due to copyright restrictions. See Fig. 3: Gibson, L. J., and M. F. Ashby. "The Mechancis
of Three-Dimensional Cellular Materials." Proceedings of The Royal Society of London A 382 (1982): 43-59.
4
Foams: Plastic Hinges
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
5
Foams: Cell Wall Fracture
Figure removed due to copyright restrictions. See Fig. 3: Gibson, L. J., and M. F. Ashby. "The Mechancis
of Three-Dimensional Cellular Materials." Proceedings of The Royal Society of London A 382 (1982): 43-59.
6
• Regardless of specific geometry chosen:
ρ∗ /ρs ∝ (t/l)2
I ∝ t4
σ ∝ F/l2
∝ δ/l
δ∝
F l3
ES I
t 4
ρ∗ 2
σ
F l
F E s t4
E ∝ ∝ 2 ∝
∝ Es
∝ Es
l δ
l F l3
l
ρs
∗
E ∗ = C1 Es (ρ∗ /ρs )2
C1 includes all geometrical constants
Data: C1 ≈ 1
• Data suggests C1 = 1
• Analysis of open cell tetrakaidecahedral cells with Plateau borders gives C1 = 0.98
• Shear modulus G∗ = C2 Es (ρ∗ /ρs )2
Isotropy: G =
C2 ∼ 3/8 if foam is isotropic
E
2(1 + ν)
• Poisson’s ratio: ν ∗ =
E
2G
ν ∗ = C3
−1=
C1
2C2
− 1 = constant, independent of Es , t/l
(analogous to honeycombs in-plane)
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Foam: Edge Bending
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
8
Poisson’s ratio
• Can make negative Poisson’s ratio foams
• Invert cell angles (analogous to honeycomb)
• Eg. thermoplastic foams - load hydrostatically
and heat to T > Tg , then cool and release load
so that edges of cell permanently point inward
Closed-cell foams
• Edge bending as for open cell foams
• Also: face stretching and gas compression
• Polymer foams: surface tension draws material to edges during processing
◦ define te , tf in figure
• Apply F to the cubic structure
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Negative Poisson’s Ratio
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
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• External work done ∝ F δ.
• Internal work bending edges ∝
Fe 2
δ e δe
∝
Es I 2
l3 δ
• Internal work stretching faces ∝ σf f vf ∝
Es f2
vf ∝ Es δ/l
2
tf l 2
δ 2
Es t4e 2
∴ F δ = α 3 δ + β Es
tf l2
l
l
F
l
E ∗ ∝ 2 → F ∝ E ∗ δl
l δ
δ 2
Es t4e 2
∗ 2
∴ E δ l = α 3 δ + β Es
tf l 2
t 4l
t l
f
e
E ∗ = αEs
+ βEs
l
l
Note: Open cells, uniform t:
ρ∗ /ρs ∝ (t/l)2
Closed cells, uniform t:
ρ∗ /ρs ∝ (t/l)
If φ is volume fraction of solid in cell edges:
te /l = Cφ1/2 (ρ∗ /ρs )1/2
tf /l = C 0 (1 − φ) (ρ∗ /ρs )
2
E∗
2
∗
= C1 φ ρ /ρs + C10 (1 − φ)ρ∗ /ρs
Es
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Closed-Cell Foam
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
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Cell Membrane Stretching
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
13
Closed cell foams - gas within cell may also contribute to E ∗
• Cubic element of foam of volume V0
• Under uniaxial stress, axial strain in direction of stress is • Deformed volume V is:
V
V0
= 1 − (1 − 2ν ∗ )
taking compressive strain as positive,
neglecting 2 , 3 terms
1 − (1 − 2ν ∗ ) − ρ∗ /ρs
Vg
=
Vg0
1 − ρ∗ /ρs
• Boyle’s law: pVg = p0 Vg0
Vg = volume gas
Vg0 = volume gas initially
p = pressure after strain p0 = pressure initially
• Pressure that must be overcome is p0 = p − p0 :
p0 (1 − 2ν ∗ )
p =
1 − (1 − 2ν ∗ ) − ρ∗ /ρs
0
• Contribution of gas compression to the modulus E ∗ :
Eg∗
dp0
p0 (1 − 2ν ∗ )
=
=
d
1 − ρ∗ /ρs
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V0 = l03
V = l1 l2 l3
1 =
l1 − l0
l0
l2 − l0
2 =
l0
→
→
l1 = l0 + 1 l0 = l0 (1 + 1 )
l2 = l0 + 2 l0
= l0 − ν1 l0
= l0 (1 − ν1 )
2
1
2 = −ν1
ν=−
3 = l0 (1 − ν1 )
V = l1 l2 l3 = l0 (1 + 1 )l0 (1 − ν1 )l0 (1 − ν1 ) = l03 (1 + 1 )(1 − ν1 )2
l03 (1 + )(1 − ν)2
V
=
= (1 + )(1 − 2ν + ν 2 2 )
3
V0
l0
= (1 − 2ν + ν 2 2 ) + − 2ν2 + ν 2 3
= 1 − + 2ν
= 1 − (1 − 2ν)
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p 0 = p − p0
p0 Vg0
p=
Vg
V 0
p0 Vg0
g
p = p − p0 =
− p0 = p0
−1
Vg
Vg
h
i
1 − ρ∗ /ρs
= p0
−1
1 − (1 − 2ν ∗ ) − ρ∗ /ρs
h 1 − ρ∗ /ρ (1 − (1 − 2ν ∗ ) − ρ∗ /ρ ) i
s
s
= p0
∗
∗
1 − (1 − 2ν ) − ρ /ρs
h
i
(1 − 2ν ∗ )
= p0
1 − (1 − 2ν ∗ ) − ρ∗ /ρs
0
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Closed cell foam
E∗
=
Es
2
φ
ρ∗
ρs
2
ρ∗
+ (1 − φ)
ρs
p0 (1 − 2ν ∗ )
+
Es (1 − ρ∗ /ρs )
↓
↓
↓
edge bending
face stretching
gas compression
• Note: if p0 = patm = 0.1 MPa, gas compression term is negligible,
except for closed-cell elastomeric foams
• Gas compression can be significant if p0 >> patm ; also modifies shape of stress plateau
in elastomeric closed-cell foams
Shear modulus: edge bending, face stretching; shear ∆V = 0 gas contribution is 0
2
∗ i
∗
3 h 2 ρ∗
ρ
= φ
+ (1 − φ)
(isotropic foam)
Es
8
ρs
ρs
Poison’s ratio = f (cell geometry only)
ν ∗ ≈ 1/3
Comparison with data
• Data for polymers, glasses, elastomers
• Es , ρs - Table 5.1 in the book
• Open cells — open symbols
• Closed cells — filled symbols
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Young’s Modulus
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
18
Shear Modulus
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
19
Poisson’s Ratio
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
20
Non-linear elasticity
Open cells:
n2 π 2 Es I
Pcr =
l2
4
Pcr
t
∗
σei ∝ 2 ∝ Es
l
l
σel∗
= C 4 Es
ρ∗
ρs
2
Data: C4 ≈ 0.05, corresponds to strain when
buckling initiates,
since
∗ 2
E ∗ = ES ρρs
Closed cells:
• tf often small compared to te (surface tension in processing) - contribution small
• If p0 >> patm , cell walls pre-tensioned, bucking stress has to overcome this
σel∗ = 0.05Es
ρ∗ 2
ρs
+ p0 − patm
• Post-collapse behavior - stress plateau rises due to gas compression (if faces don’t rupture) ν ∗ = 0
in post-collapse regime
∗ 2
∗
p
(1
−
2ν
)
p
p0 ρ
0
0
∗
p0 =
=
σ
=
0.05
E
+
s
p
ost-collapse
1 − (1 − 2ν ∗ ) − ρ∗ /ρs
1 − − ρ∗ /ρs
ρs
1 − − ρ∗ /ρs
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Elastic Collapse Stress
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
22
Elastic Collapse Stress
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
23
Post-collapse stress strain curve
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
24
Plastic collapse
Open cells:
• Failure when M = Mp
• Mp ∝ σys t3
∗ 3
M ∝ σpl
l
∗
σpl
= C5 σys (ρ∗ /ρs )3/2
C5 ∼ 0.3 from data.
∗
• Elastic collapse precedes plastic collapse if σel∗ < σpl
• 0.05 Es (ρ∗ /ρs )2 ≤ 0.3 σys (ρ∗ /ρs )3/2
(ρ/ρs )critical
≤ 36 (σys /Es )2
σ
1
rigid polymers (ρ∗ /ρs )cr < 0.04 Eyss ∼ 30
σ
1
metals (ρ∗ /ρs )cr < 10−5 Eyss ∼ 1000
Closed cells:
• Including all terms:
∗
σpl
∗ 3/2
∗
ρ
ρ
0
= C5 σys φ
+ C5 σys (1 − φ)
+ p0 − patm
ρs
ρs
↑
↑
↑
gas
edge bending
face stretching
∗
∗
• But in practice, faces often rupture around σpl
- often σpl
= 0.3 (ρ∗ /ρs )3/2 σys
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Plastic Collapse Stress
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
26
Plastic Collapse Stress
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
27
Brittle crushing strength
Open cells:
• Failure when M = Mf
∗ 2
M ∝ σcr
l
σcr = C6 σfs (ρ∗ /ρs )3/2
C6 ≈ 0.2
Mf ∝ σfs t3
Densification strain, D :
• At large comp. strain, cell walls begin to touch, σ − rises steeply
• E ∗ → Es ; σ − curve looks vertical, at limiting strain
• Might expect D = 1 − ρ∗ /ρs
• Walls jam together at slightly smaller strain than this:
D = 1 − 1.4 ρ∗ /ρs
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Densification Strain
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
29
Fracture toughness
Open cells: crack length 2a, local stress σl , remote stress σ ∞
√
Cσ ∞ πa
σl = √
2πr
a distance r from crack tip
• Next unbroken cell wall a distance r ≈ l/2, a head of crack tip subject to a force (integrating stress
over next cell)
F ∝ σl l 2 ∝ σ ∞
pa
l
l2
• Edges fail when applied moment, M = fracture moment, Mf
Mf = σfs t3
∞
a 1/2
l
M ∝ Fl∝σ
l
l 1/2 t 3
∞
σ ∝ σfs
a
l
∗
KIC
= σ∞
√
3
M = Mf
√
πa = C8 σfs
πl ρ∗ /ρs
3/2
→
σ
∞
a 1/2
l
l3 ∝ σfs t3
Data: C8 ∼ 0.65
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Fracture Toughness
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
31
Fracture Toughness
Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge
University Press, © 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.
32
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