Lecture 4 Honeycombs Notes, 3.054

Transcription

1 Honeycombs-In-plane behavior Lecture 4 Honeycombs Notes, 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 cell walls touch Increasing t/l E σ E D 1

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

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

4 Deformation mechanisms Bending X 1 Loading Bending X 2 Loading Bending Shear Buckling Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge University Press, Figure courtesy of Lorna Gibson and Cambridge University Press. 4

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

6 Stress-Strain Curve Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge University Press, Figure courtesy of Lorna Gibson and Cambridge University Press. 6

7 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 ρ ( h l l + 2) 2 t = = ρ h s 2 cos θ ( sin θ) 3 l regular hexagons Solid cell wall properties: ρ s,e s,σ ys,σ fs Cell geometry: h/l, θ l

8 In-plane properties Assumptions: t/l small ((ρ /ρ s ) small) neglect axial and shear contribution to deformation c 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 E 1 1/E 1 ν 21 /E 2 ν 31 /E σ 1 E 2 ν 12 /E 1 1/E 2 ν 32 /E σ 2 E 3 ν 13 /E 1 ν 23 /E 2 1/E σ 3 E = /G σ 4 E /G 13 0 σ 5 E /G 12 σ 6 8

9 Matrix notation: E 1 = E 11 E 4 = γ 23 σ 1 = σ 11 σ 4 = σ 23 E 2 = E 22 E 5 = γ 13 σ 2 = σ 22 σ 5 = σ 13 E 3 = E 33 E 6 = γ 12 σ 3 = σ 33 σ 6 = σ 12 In plane (x 1 x 2 ): 4 independent elastic constants: E 1 E 2 ν 12 G 12 ν 12 ν 21 and compliance matrix symmetric = (reciprocal relation) E 1 E 2 [ E j notation for Poisson s ratio: ν ij = ] E i Young s modulus in x 1 direction P σ 1 = (n + l sin θ) b Unit cell in x 1 direction: 2l cos θ Unit cell in x 2 direction: h +2l sin θ δ sin θ E 1 = l cos θ 9

10 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

11 M diagram: 2 cantilevers of length l/2 P sin θ(l/2) 3 δ = 2 3E s I 2 P l 3 sin θ = 24 E s I P l 3 sin θ b t 3 δ = I = 12 E s I 12 Combining: σ 1 P l cos θ E 1 = = E 1 (h + l sin θ) b δ sin θ P l cos θ b t 3 = 12 Es (h + l sin θ) b P l 3 sin 2 θ 12 ( t ) 3 cos θ 4 ( t ) E 3 1 = E s = E l (h/l + sin θ) sin 2 θ 3 l solid relative cell geometry property density s regular hexagons h/l=1 θ = 30 11

12 Poisson s ratio for loading in x 1 direction E ν = 2 12 E 1 δ sin θ δ cos θ E 1 = E 2 = (lengthens) l cos θ ( h + lsin θ δ cos θ l cos θ cos 2 θ ν 12 = = h + l sin θ δsin θ (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 3/4 (3/2)( 1/2) e.g. h/l=2 θ = 30 ν 12 = = 1 E 2 ν 12 G 12 Can be found in similar way; results in book 12

13 Compressive strength (plateau stress) Cell collapse by: (1) elastic buckling (2) plastic yielding (3) brittle crushing peaks and valleys correspond to fracture of indi vidual cell walls buckling of vertical struts throughout honeycomb localization of yield as deformation progresses, propagation of failure band Plateau stress: elastic buckling, σ e l Elastomeric honeycombs cell collapse by elastic buckling of walls of length h when loaded in x 2 direction No buckling for σ 1 ; bending of inclined walls goes to densification n=end constraint factor Euler buckling load pin pin fixed fixed n 2 π 2 E s I P cr = n=1 n=2 h 2 13

14 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

15 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 n/l= n= P cr n 2 π 2 E s bt 3 and (σ el ) 2 = = 2l cos θ b h 2 2l cos θ b 12 (σ el ) 2 = n 2 π 2 (t/l) 3 E s 24 (h/l) 2 cos θ regular hexagons: (σ ) 2 = 0.22 E s (t/l) 3 el and since E 2 = 4/ 3 E s (t/l) 3 = 2.31 E s (t/l) 3 strain at buckling (E el) 2 = 0.10, for regular hexagons, independent of E s, t/l 15

16 Plateau stress: plastic yielding, σ pl Failure by yielding in cell walls Yield strength of cell walls = σ ys Plastic hinge forms when cross section fully yields Beam theory linear elastic σ = My I Once stress outer fiber=σ ys, 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

17 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

18 Moment at for mat ion of plastic hinge (plastic moment, M p ): ( bt )( t ) σ ys bt 2 M p = σ ys = Applied moment, from applied stress 2M app PL sin θ = 0 Pl sin θ M app = 2 P σ1 = (h + l sin θ) b M app = σ 1 (h + l sin θ) b l sin θ 2 Plastic collapse of honeycomb at (σ pl ) 1, when M app = M p l sin θ bt 2 ( ) σ pl ) 1 (h + l sin θ) )b) = σ ys 2 42 ( t ) 2 1 (σ pl ) 1 = σ ys l 2(h/l + sin θ) sin θ 2 ( t ) regular hexagons: (σ pl ) 1 = σ ys ( ) 3 l similarly, (σ t pl ) 2 = σ 2 1 ys l 2 cos 2 θ 2 18

19 For thin-walled honeycombs, elastic buckling can precede plastic collapse ( for σ 2 ) Elastic buckling stress = plastic collapse stress (σ el ) 2 = (σ pl ) 2 n 2 π 2 (t/l) 3 σ ys (t/l) 2 E s = 24 (h/l) 2 cos θ 2 cos 2 θ 12 (h/l) 2 ( σ ) ys (t/l) critical = n 2 π 2 cos θ E s σ ys regular hexagons:(t/l) critical = 3 Es E.g. metals σ ys /E s.002 (t/l) critical 0.6% most metal honeycomb denser than this polymer σ ys /E s 3 5% (t/l) critical 10-15% low density polymers with yield point may buckle before yield 19

20 Plastic stress: brittle crushing, (σ cr ) 1 Ceramic honeycombs fail in brittle manner Cell wall bending stress reaches modulus of rapture wall fracture loading in x 1 direction: P = σ 1 (h + l sin θ) b σ fs = modulus of rupture of cell wall Plsin θ σ 1 (h + l sin θ) blsin θ M max. applied = = 2 2 Moment at fracture, M f ( 1 t )( 2 ) M f = σ f s b t = σfs bt ( t ) 2 1 (σ cr ) 1 = σ fs l 3(h/l + sin θ) sin θ regular hexagons: (σ c r ) 1 = 4 ( t ) σ 2 fs 9 l 20

21 Brittle Crushing Gibson, L. J., and M. F. Ashby. Cellular Solids: Structure and Properties. 2nd ed. Cambridge University Press, Figure courtesy of Lorna Gibson and Cambridge University Press. 21

22 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, σ fs Continuum: crack of length 2c in a linear elastic solid material normal to a remote tension stress σ 1 creates a local stress field at the crack tip σ 1 πc σ local = σ l = 2πr 22

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

24 Honeycomb: cell walls bent fail when applied moment = fracture moment M app P l on wall A σ 1 c l 2 b M app P l σ l l 2 b σ fs b t 2 l ( t ) 2 l (σ f ) 1 σ fs l c ( t ) 2 K I C = σ f πc = c σ fs l depends on cell size, l! l c=constant Summary: hexagonal honeycombs, in-plane properties Linear elastic moduli: E G 1 E 2 ν Plateau stresses (σ el ) 2 elastic buckling (compression) σ pl plastic yield σ brittle crushing Fracture toughness (tension) cr K IC brittle fracture 24

25 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% depth b into page Force in each member proportional to P P δ Pl σ ɛ δ axial shortening: Hooke s law lb l AEs σ P l P bte ( s t ) E E s ɛ lb δ b Pl l E = ce s (t/l) exact calculation: E =1.15 E s (t/l) for equilateral triangles 25

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

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