Lecture 4 Honeycombs Notes, 3

Lecture 4 Honeycombs Notes, 3.054Honeycombs-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-PlaneCompression 3 regimes– linear elastic bending– stress plateau buckling yielding brittle crushing– densification Increasing t/l E σ cell walls touchED 1

Honeycomb GeometryGibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. CambridgeUniversity 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. CambridgeUniversity Press, 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.3

DeformationmechanismsBendingX1 LoadingBendingX2 LoadingBendingShearBucklingGibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. CambridgeUniversity Press, 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.4

Plastic collapse in analuminum honeycombGibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. CambridgeUniversity Press, 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.5

Stress-Strain CurveGibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. CambridgeUniversity 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 tensionVariables affecting honeycomb properties Relative densitythl ( l 2)2 tρ ρs2 cos θ ( hl sin θ)3 l Solid cell wall properties: ρs , Es , σys , σf s Cell geometry: h/l, θregular hexagons

In-plane propertiesAssumptions: t/l small ((ρ c /ρs ) small) — neglect axial and shear contribution to deformation Deformations small — neglect changes in geometry Cell wall — linear elastic, isotropicSymmetry Honeycombs are orthotropic — rotate 180 about each of three mutually perpendicular axes andstructure is the sameLinear elastic deformation 1/E1 ν21 /E2 ν31 /E2000σ1E1 E2 ν12 /E1 1/E2 ν32 /E3 000 σ2 σ3 E3 ν13 /E1 ν23 /E2 1/E3000 σ4 E4 0001/G0023 E5 0 σ5 00001/G13E6000001/G12σ68

Matrix notation:E1 E11 E4 γ23 σ1 σ11 σ4 σ23E2 E22 E5 γ13 σ2 σ22 σ5 σ13E3 E33 E6 γ12 σ3 σ33 σ6 σ12 In plane (x1 x2 ): 4 independent elastic constants:E1 E2 ν12 G12and compliance matrix symmetric ν12 ν21 E2E1(reciprocal relation)h Ej inotation for Poisson’s ratio: νij EiYoung’s modulus in x1 directionσ1 Unit cell in x1 direction: 2l cos θUnit cell in x2 direction: h 2l sin θP(n l sin θ) bE1 9δ sin θl cos θ

In-Plane Deformation:Linear ElasticityFigure 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/2P sin θ(l/2)3δ 2·3Es I32 P l sin θ 24 Es Ib t3P l3 sin θδ I 1212 Es ICombining:σ1Pl cos θ E1(h l sin θ) b δ sin θb t3Pl cos θ 12 Es12(h l sin θ) b P l3 sin2 θ t 3cos θ sin θ) bMmax.applied σf s modulus of rupture of cell wallP l sin θσ1 (h l sin θ) b l sin θ 22Moment at fracture, Mft 2 σf s b t2Mf t σf s b22 36 1(σc r )1 σf s t 2lregular hexagons:13 (h/l sin θ) sin θ(σc r )120 t 24 σf s9l

Brittle CrushingGibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. CambridgeUniversity 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 fractureFracture toughnessAssume: crack length large relative to cell size (continuum assumption) axial forces can be neglected cell wall material has constant modulus of rapture, σf sContinuum: crack of length 2c in a linear elastic solid material normal to a remote tension stressσ1 creates a local stress field at the crack tipσlocal σ1 π c σl 2π r22

Fracture ToughnessGibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. CambridgeUniversity Press, 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.23

Honeycomb: cell walls bent — fail when applied moment fracture momentMapp P lon wall A 2σ1 cl b Mapp P l σl l2 b σ f s b t2l t 2 l (σf )1 σf slc t 2 depends on cell size, l!KI C σf πc c σf sllc constantSummary: hexagonal honeycombs, in-plane properties Linear elastic moduli: Plateau stresses(compression) Fracture toughness(tension)E1 (σel )2 σpl σcr KICE2 ν12 G12elastic bucklingplastic yieldbrittle crushingbrittle fracture24

Honeycombs: In-plane behavior — triangular cells Triangulated structures trusses Can analyze as pin jointed (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 Pdepth b into pagePδPl δ axial shortening: Hooke’s lawlblA Es t σP lP b t Es E Es lb δb Pllσ E c Es (t/l)exact calculation: E 1.15 Es (t/l) for equilateral triangles25

Square and TriangularHoneycombsGibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. CambridgeUniversity Press, 1997. Figure courtesy of Lorna Gibson and Cambridge University Press.26

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2 θ ν 12 h l sin θδ sin θ l (h/l sin θ) sin θ ν 12 depends ONLY on cell geometry (h/l, θ), not on E s, t/l Regular hexagonal cells: ν 12 1 ν can be negative for θ 0 h/l 2 θ 30 e.g. ν 3/4 12 (3/2)( 1/2) 1 νE 2