Lecture 7, Foams, 3.054

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 t 2, 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 3

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 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 /ρ s (t/l) 2 I t 4 σ F/l 2 F l 3 ɛ δ/l δ ES I σ F l F E E ɛ l2 δ st 4 ( t ) 4 ρ ) 2 l F l E 3 s E s l (ρ s E = C 1 E s (ρ /ρ s ) 2 Data suggests C 1 = 1 C 1 includes all geometrical constants Data: C 1 1 Analysis of open cell tetrakaidecahedral cells with Plateau borders gives C 1 = 0.98 Shear modulus G = C 2 E s (ρ /ρ s ) 2 C 2 3/8 if foam is isotropic E Isotropy: G = 2(1 + ν) Poisson s ratio: ν = E 2G 1 = C 1 2C 2 1 = constant, independent of E s, t/l ν = C 3 (analogous to honeycombs in-plane) 7

Foam: Edge Bending Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 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 > T g, 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 t e, t f in figure Apply F to the cubic structure 9

Negative Poisson s Ratio Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 10

External work done F δ. Internal work bending edges F e δ e δ 2 e E si δ 2 l 3 ) 2 Internal work stretching faces σ f ɛ f v f E s ɛf 2 v f E 2 s (δ/l t f l Est 4 ( e δ ) 2tf F δ = α l 3 δ2 + β E s l 2 l E F l F l 2 δ E δl E δ 2 E s t 4 ( e δ ) 2tf l = α l 3 δ2 + β E s l 2 ( l E te ) 4 ( tf ) = αe s + βe s l l Note: Open cells, uniform t: If φ is volume fraction of solid in cell edges: ρ /ρ s (t/l) 2 t e/l = Cφ 1/2 ( ρ /ρ s ) 1/2 Closed cells, uniform t: t f /l = C (1 φ)(ρ /ρ s ) ρ /ρ s (t/l) ( ) = C 1 φ 2 ρ /ρ s E s E 2 + C 1 (1 φ)ρ /ρ s 11

Closed-Cell Foam Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 12

Cell Membrane Stretching Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 13

Closed cell foams - gas within cell may also contribute to E Cubic element of foam of volume V 0 Under uniaxial stress, axial strain in direction of stress is ɛ Deformed volume V is: V V 0 = 1 ɛ(1 2ν ) taking compressive strain as positive, neglecting ɛ 2, ɛ 3 terms V g = 1 ɛ(1 2ν ) ρ /ρ s V v Vg 0 ρ g = olume gas 1 /ρ s Vg 0 = volume gas initially Boyle s law: pv g = p 0 V 0 g p = pressure after strain ɛ p 0 = pressure initially Pressure that must be overcome is p = p p 0 : p p 0 ɛ (1 2ν ) = 1 ɛ (1 2ν ) ρ /ρ s Contribution of gas compression to the modulus E : dp E g = dɛ = p 0(1 2ν ) 1 ρ /ρ s 14

l V 0 = l 3 1 0 ɛ 1 = l 0 l 0 l 1 = l 0 + ɛ 1 l 0 = l 0 (1 + ɛ 1 ) l 2 V = l 1 l 2 l 3 ɛ 2 = l ɛ 0 l 2 2 = l 0 + ɛ 2 l 0 ν = ɛ 1 l 0 = l 0 νɛ 1 l 0 ɛ 2 = νɛ 1 = l 0 (1 νɛ 1 ) ɛ 3 = l 0 (1 νɛ 1 ) 3 2 V = l 1 l2l 3 = l 0 (1 + ɛ 1 )l 0 (1 νɛ 1 )l 0 (1 νɛ 1 ) = l 0 (1 + ɛ 1 )(1 νɛ 1 ) V = l3 0(1 + ɛ)(1 νɛ) 2 2 = (1 + ɛ)(1 V 3 2νɛ + ν ɛ 2 ) 0 l0 = (1 2νɛ + ν 2 ɛ 2 ) + ɛ 2νɛ 2 + ν 2 ɛ 3 = 1 ɛ + 2νɛ = 1 ɛ(1 2ν) 15

p = p p 0 p 0 V g p = Vg p p 0V ( 0 g V ) g = p p 0 = p 0 = p 0 1 V g V g [ 1 ρ /ρ ] s = p 0 1 1 ɛ(1 2ν ) ρ /ρ s [ 1 ρ /ρ s (1 ɛ(1 2ν ) ρ /ρ s ) ] = p 0 1 ɛ(1 2ν ) ρ /ρ s [ ɛ(1 2ν ) ] = p 0 1 ɛ(1 2ν ) ρ /ρ s 16

Closed cell foam E E s = ( ρ ) 2 ( ) ρ φ 2 p0(1 2ν ) + (1 φ) + ρ s ρ s E s (1 ρ /ρ s ) edge bending face stretching gas compression Note: if p 0 = p atm = 0.1 MPa, gas compression term is negligible, except for closed-cell elastomeric foams Gas compression can be significant if p 0 >> p atm ; also modifies shape of stress plateau in elastomeric closed-cell foams Shear modulus: edge bending, face stretching; shear V = 0 gas contribution is 0 ɛ = 3 ( ) ρ [φ 2 2 ) ρ ] + (1 φ)( (isotropic foam) E s 8 Poison s ratio = f (cell geometry only) ν 1/3 Comparison with data Data for polymers, glasses, elastomers E s, ρ s - Table 5.1 in the book Open cells open symbols Closed cells filled symbols ρ s ρ s 17

Young s Modulus Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 18

Shear Modulus Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 19

Poisson s Ratio Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 20

Non-linear elasticity Open cells: n 2 π 2 E s I P cr = l 2 σei P ( ) 4 ( cr t E l ρ 2 s σ l el = C 4 E s Closed cells: ρ s ) 2 Data: C 4 0.05, corresponds to strain when buckling ( initiates, since E = E ρ ) 2 S ρ s t f often small compared to t e (surface tension in processing) - contribution small If p 0 >> p atm, cell walls pre-tensioned, bucking stress has to overcome this σ el ( = 0.05E ρ ) 2 s + p0 p ρ s atm Post-collapse behavior - stress plateau rises due to gas compression (if faces don t rupture) ν = 0 in post-collapse regime p p 0 ɛ(1 2ν ) = = 1 ɛ(1 2ν ) ρ /ρ s ( p 0 ɛ ρ ) 2 σ 1 ɛ ρ /ρ post-collapse = 0.05 E s + s ρ s p 0 ɛ 1 ɛ ρ /ρ s 21

Elastic Collapse Stress Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 22

Elastic Collapse Stress Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 23

Post-collapse stress strain curve Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 24

Plastic collapse Open cells: Failure when M = M p M p σ ys t 3 M σ pl l 3 σ pl = C 5 σ ys (ρ /ρ s ) 3/2 C 5 0.3 from data. Elastic collapse precedes plastic collapse if σ el < σ pl 0.05 E s(ρ /ρ s) 2 0.3 σ 3 ys (ρ /ρ s ) /2 σ rigid polymers (ρ /ρ s ) < 0.04 ( ) ys cr E s 1 30 (ρ/ρ s ) critical 36 (σ ys /E s ) 2 metals (ρ /ρ s ) cr < 10 ( 5 σ ys ) E s 1 1000 Closed cells: ( ) ρ Including all terms: σ 3 /2 ( ) ρ pl = C 5 σ ys φ + C 5 σ ys(1 φ) + ρ ρ s edge bending face stretching s p 0 p gas 3 But in practice, faces often rupture around σpl - often σ pl = 0.3 (ρ /ρ s ) /2 σ ys atm 25

Plastic Collapse Stress Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 26

Plastic Collapse Stress Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 27

Brittle crushing strength Open cells: Failure when M = M f M σ cr l 2 M f σ fs t 3 σ cr = C 6 σ fs (ρ /ρ s ) 3/2 C 6 0.2 Densification strain, ɛ D : At large comp. strain, cell walls begin to touch, σ ɛ rises steeply E E s ; σ ɛ curve looks vertical, at limiting strain Might expect ɛ D = 1 ρ /ρ s Walls jam together at slightly smaller strain than this: ɛ D = 1 1.4 ρ /ρ s 28

Densification Strain Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 29

Fracture toughness Open cells: crack length 2a, local stress σ l, remote stress σ σ l = Cσ πa 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 σ l l 2 σ a l 2 l Edges fail when applied moment, M = fracture moment, M f M f = σ fs t 3 ( a M F l σ l ( σ l ) 1/2 ( t ) 3 σ fs a l ) 1/2 l 3 M = M f σ ( a l ) 1/2 l 3 σ fs t 3 K IC = σ πa = C 8 σ fs πl ( ρ /ρ s ) 3/2 Data: C 8 0.65 30

Fracture Toughness Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 31

Fracture Toughness Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge 32

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