Lecture 4, Honeycombs Notes, 3.054

Lecture 4 Honeycombs Notes, 3.054 Honeycombs-In-plane behavior • Prismatic cells • Polymer, metal, ceramic honeycombs widely available • Used for sandwich structure cores, energy absorption, carriers for catalysts • Some natural materials (e.g. wood, cork) can be idealized as honeycombs • Mechanisms of deformation and failure in hexagonal honeycombs parallel those in foams ◦ simpler geometry — unit cell — easier to analyze • Mechanisms of deformation in triangular honeycombs parallel those in 3D trusses (lattice materials) Stress-strain curves and Deformation behavior: In-Plane Compression • 3 regimes – linear elastic − bending – stress plateau − buckling − yielding − brittle crushing – densification • Increasing t/l ⇒ E ∗ ↑ σ∗ ↑ − cell walls touch ED ↓ 1 Honeycomb Geometry 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. 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. 3 Deformation mechanisms Bending X1 Loading Bending X2 Loading Bending Shear Buckling 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. 4 Plastic collapse in an aluminum honeycomb 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 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. 6 Tension • Linear elastic � bending • Stress plateau � exists only if cell walls yield � no buckling in tension � brittle honeycombs fracture in tension Variables affecting honeycomb properties � � • Relative density t h l ( l + 2) 2 t ρ∗ √ = = ρs 2 cos θ ( hl sin θ) 3 l • Solid cell wall properties: ρs , Es , σys , σf s • Cell geometry: h/l, θ regular hexagons In-plane properties Assumptions: • t/l small ((ρ∗c /ρs ) small) — neglect axial and shear contribution to deformation • Deformations small — neglect changes in geometry • Cell wall — linear elastic, isotropic Symmetry • Honeycombs are orthotropic — rotate 180◦ about each of three mutually perpendicular axes and structure is the same Linear elastic deformation ⎡ ⎤ ⎡ ⎤⎡ ⎤ 1/E1 −ν21 /E2 −ν31 /E2 0 0 0 σ1 E1 ⎢E2 ⎥ ⎢−ν12 /E1 1/E2 −ν32 /E3 ⎥ ⎢ 0 0 0 ⎥ ⎢σ2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢σ3 ⎥ ⎢E3 ⎥ ⎢−ν13 /E1 −ν23 /E2 −1/E3 0 0 0 ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎥ ⎢ σ4 ⎥ ⎢E4 ⎥ ⎢ 0 0 0 1/G 0 0 23 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎣E5 ⎦ ⎣ 0 ⎦ ⎣ σ5 ⎦ 0 0 0 0 1/G13 E6 0 0 0 0 0 1/G12 σ6 8 • Matrix notation: E1 = E11 E4 = γ23 σ1 = σ11 σ4 = σ23 E2 = E22 E5 = γ13 σ2 = σ22 σ5 = σ13 E3 = E33 E6 = γ12 σ3 = σ33 σ6 = σ12 • In­plane (x1 � x2 ): 4 independent elastic constants: E1 E2 ν12 G12 and compliance matrix symmetric �ν12 �ν21 = E2 E1 (reciprocal relation) h �Ej i notation for Poisson’s ratio: νij = Ei Young’s modulus in x1 direction σ1 = Unit cell in x1 direction: 2l cos θ Unit cell in x2 direction: h + 2l sin θ P (n + l sin θ) b E1 = 9 δ sin θ l cos θ In-Plane Deformation: Linear Elasticity Figure removed due to copyright restrictions. See Figure 5: L. J. Gibson, M. F. Ashby, et al. "The Mechanics of Two-Dimensional Cellular Materials." 10 M diagram: 2 cantilevers of length l/2 P sin θ(l/2)3 δ =2· 3Es I 3 2 P l sin θ = 24 Es I b t3 P l3 sin θ δ= I= 12 12 Es I Combining: σ1 P l cos θ = E1 (h + l sin θ) b δ sin θ b t3 P l cos θ = 12 Es 12 (h + l sin θ) b P l3 sin2 θ  t 3 cos θ 4  t 3 ∗ =√ Es E1 = Es l (h/l + sin θ) sin2 θ 3 l E1∗ = ↑ ↑ solid relative property density ↑ cell geometry 11 regular hexagons h/l=1 θ = 30◦ Poisson’s ratio for loading in x1 direction ∗ =� ν12 E2 E1 δ sin θ δ cos θ E2 = (lengthens) l cos θ h + llsin θ δ cos θ l cos θ cos2 θ = = h + l sin θ δ sin θ (h/l + sin θ) sin θ E1 = ∗ ν12 ∗ • ν12 depends ONLY on cell geometry (h/l, θ), not on Es , t/l ∗ =1 • Regular hexagonal cells: ν12 • ν can be negative for θ < 0 e.g. h/l=2 E∗2 ∗ ν12 θ = �30◦ ∗ ν12 = 3/4 (3/2)(�1/2) = �1 G∗12 • Can be found in similar way; results in book 12 Compressive strength (plateau stress) • Cell collapse by: (1) elastic buckling (2) plastic yielding • buckling of vertical struts throughout honeycomb (3) brittle crushing • localization of yield • as deformation progresses, propagation of failure band • peaks and valleys corre­ spond to fracture of indi­ vidual cell walls Plateau stress: elastic buckling, σe∗l • Elastomeric honeycombs — cell collapse by elastic buckling of walls of length h when loaded in x2 direction • No buckling for σ1 ; bending of inclined walls goes to densification n=end constraint factor Euler buckling load pin­pin n=1 n2 π 2 Es I Pcr = h2 13 fixed­fixed n=2 Elastic Buckling Figure removed due to copyright restrictions. See Figure 7: L. J. Gibson, M. F. Ashby, et al. "The Mechanics of Two-Dimensional Cellular Materials." 14 • Here, constraint n depends on stiffness of adjacent inclined members • Can find elastic line analysis (see appendix if interested) • Rotational stiffness at ends of column, h, matched to rotational stiffness of inclined members • Find and n/l=1 n=0.686 1.5 0.760 2 0.860 (σel∗ )2 n2 π 2 Es bt3 Pcr = = 2l cos θ b h2 2l cos θ b 12 (σel∗ )2 (t/l)3 n2 π 2 = Es 24 (h/l)2 cos θ regular hexagons: and since strain at buckling (σel∗ )2 = √ 0.22 Es (t/l)3 E2∗ = 4/ 3 Es (t/l)3 = 2.31 Es (t/l)3 (E∗el )2 = 0.10, for regular hexagons, independent of Es , t/l 15 ∗ Plateau stress: plastic yielding, σpl • Failure by yielding in cell walls • Yield strength of cell walls = σys • Plastic hinge forms when cross­section fully yields • Beam theory — linear elastic σ = My I • Once stress outer fiber=σys , yielding begins and then progresses through the section, as the load increases as P ↑ • When section fully yielded (right figure), form plastic “hinge” • Section rotates like a pin 16 Plastic Collapse Figure removed due to copyright restrictions. See Figure 8: L. J. Gibson, M. F. Ashby, et al."The Mechanics of Two-Dimensional Cellular Materials." 17 • Moment at formation of plastic hinge (plastic moment, Mp ):  b t  t  σys b t2 Mp = σys = 2 2 4 • Applied moment, from applied stress 2Mapp � P L sin θ = 0 P l sin θ 2 P σ1 = (h + l sin θ) b Mapp = Mapp = σ1 (h + l sin θ) b ∗ )1 , when Mapp = Mp • Plastic collapse of honeycomb at (σpl l sin θ b) t2 (h + l sin θ) )b) = σys 2 42  t 2 1 ∗ (σpl )1 = σys l 2(h/l + sin θ) sin θ ∗ (σpl )1  t 2 2 regular hexagons: = σys 3 l  t 2 1 ∗ similarly, (σpl )2 = σys l 2 cos2 θ ∗ (σpl )1 18 l sin θ 2 • For thin-walled honeycombs, elastic buckling can precede plastic collapse ( for σ2 ) ∗ • Elastic buckling stress = plastic collapse stress (σel∗ )2 = (σpl )2 n2 π 2 (t/l)3 σys (t/l)2 Es = 2 cos2 θ 24 (h/l)2 cos θ (t/l)critical 12 (h/l)2  σys  = 2 2 n π cos θ Es regular hexagons:(t/l)critical = 3 • E.g. metals σys /Es ∼ .002 σys Es (t/l)critical ∼ 0.6% ◦ most metal honeycomb denser than this polymer σys /Es ∼ 3 − 5% (t/l)critical ∼ 10-15% ◦ low density polymers with yield point may buckle before yield 19 ∗ Plastic stress: brittle crushing, (σcr )1 • Ceramic honeycombs — fail in brittle manner • Cell wall bending — stress reaches modulus of rapture — wall fracture loading in x1 direction: P = σ1 (h + l sin θ) b Mmax. applied = σf s = modulus of rupture of cell wall P l sin θ σ1 (h + l sin θ) b l sin θ = 2 2 Moment at fracture, Mf t  2  σf s b t2 Mf = t = σf s b 2 2 3 6 1 (σc∗r )1 = σf s  t 2 l regular hexagons: 1 3 (h/l + sin θ) sin θ (σc∗r )1 20  t 2 4 = σf s 9 l Brittle Crushing 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. 21 Tension • No elastic buckling • Plastic plateau stress approx. same in tension and compression (small geometric difference due to deformation) • Brittle honeycombs: fast fracture Fracture toughness Assume: • crack length large relative to cell size (continuum assumption) • axial forces can be neglected • cell wall material has constant modulus of rapture, σf s Continuum: crack of length 2c in a linear elastic solid material normal to a remote tension stress σ1 creates a local stress field at the crack tip σlocal √ σ1 π c = σl = √ 2π r 22 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. 23 Honeycomb: cell walls bent — fail when applied moment = fracture moment Mapp ∝ P l on wall A √ 2 σ 1 cl b √ Mapp ∝ P l ∝ σl l2 b ∝ ∝ σ f s b t2 l �  t 2 l ∗ (σf )1 ∝ σf s l c  t 2 √ √ ∗ ∗ depends on cell size, l! KI C = σf πc = c σf s l l c=constant Summary: hexagonal honeycombs, in-plane properties • Linear elastic moduli: • Plateau stresses (compression) • Fracture toughness (tension) E1∗ (σel∗ )2 ∗ σpl ∗ σcr ∗ KIC E2∗ ∗ ν12 ∗ G12 elastic buckling plastic yield brittle crushing brittle fracture 24 Honeycombs: In-plane behavior — triangular cells • Triangulated structures ­ trusses • Can analyze as pin­jointed (no moment at joints) • Forces in members all axial (no bending) • If joints fixed and include bending, difference ∼ 2% • Force in each member proportional to P depth b into page P δ Pl ∝ δ∝ axial shortening: Hooke’s law lb l A Es t σ P l P b t Es ∗ E ∝ ∝ ∝ ∝ Es  lb δ b Pl l σ∝ E ∗ = c Es (t/l) exact calculation: E ∗ = 1.15 Es (t/l) for equilateral triangles 25 Square and Triangular Honeycombs 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