Mechanically Efficient Cellular Microstructures in Plants. Lorna J. Gibson Materials Science & Engineering MIT

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1 Mechanically Efficient Cellular Microstructures in Plants Lorna J. Gibson Materials Science & Engineering MIT 1

2 Introduction Plants are typically loaded in bending by wind and in compression by self-weight Minimizing mass reduces metabolic cost to grow material Examine strategies used in plants to reduce mass 2

3 Introduction Wood: Uniform honeycomb-like structure Palm stem: Radial density gradient Plant stem: Cylindrical shell with compliant core Monocotyledon leaves: Sandwich structures 3

4 Wood: Honeycomb-Like Microstructure Cedar 4

5 Wood: Honeycomb Models E * E salong = ρ* ρ s Cell wall: Fiber Composite Model E * = ρ* E sacross ρ s 3 5

6 Wood in Bending: E 1/2 /ρ ( E * ) 1/2 ρ * ( = E s) 1/2 ρ s ρ s ρ * 1/2 Wood cell wall Stiffness performance index for wood in bending is similar to that for best engineering composites 6

7 Wood in Bending: σ f 2/3 /ρ ( * σ ) 2/3 f ρ * ( = σ ) 2/3 ys ρ s ρ s ρ * 1/3 Strength performance index for wood in bending is similar to that for best engng composites 7

8 Wood Tree in bending loaded as cantilever Radius decreases with distance away from the ground Further increases mechanical performance of the tree E = constant, r = r (z) 8

9 Palm Stem: Radial Density Gradient (Also Bamboo) 9

Palm Stem: A Different Strategy Stem has constant diameter: r = constant As palm grows taller, it increases the density of the material towards its periphery Cell

10 Palm Stem: A Different Strategy Stem has constant diameter: r = constant As palm grows taller, it increases the density of the material towards its periphery Cell wall thickness increases towards periphery of stem and towards the base of the stem E = E (r, z) Coconut Palm Image:Palmtree_Curacao.jpg 10

11 Palm: Microstructure of Peripheral Stem Tissue Rich, μm 10 μm Kuo-Huang et al., 2004 Young Old 11

12 Palm Stem: Density Gradient Rich, PM (1987) Bot.Gazette 148,

13 Palm Stem: Density at Breast Height Densities of common woods A single mature palm has a similar range of density as nearly all species of wood combined Rich, PM (1987) Bot.Gazette 148,

14 Palm Stem: Density Gradient 14

15 Palm Stem: Mechanical Properties vs. Density Rich, PM (1987) Bot.Gazette 148,

16 Density Gradient: Iriartea gigantea ρ = r r o n ρ max ( EI) gradient = Cπr o mn E = C ρ ρ max m = C r r o mn ( EI) gradient = ( EI) uniform 4 mn + 4 n m Iriartea palm: n = 2, m = 2.5, (EI) gradient /(EI) uniform = 2.5 Similar calculation for Welfia georgii, gives (EI) gradient /(EI) uniform =

17 Palm Stem: Bending Stress Distribution σ(y) = Eε = Eκ y σ(r,θ) = C r r o mn κr cosθ r mn+1 Iriartea gigantea: m = 2.5, n =2 σ r 6 17

18 Palm Stem: Bending Strength Distribution σ * ρ ρ max q r r o nq Iriartea gigantea: n = 2, q = 2 σ * r 4 Strength matches bending stress distribution 18

19 Plant Stems: Cylindrical Shells with Compliant Cores (Also in Animal Quills, Toucan Beak) 19

20 Plant Stems Milkweed Grassy stem 20

21 Milkweed Stem 21

22 Grassy Stem Hollow struts 22

23 Plant stems Circular tube cross-section Resists bending (wind loads) Maximize shape factor Φ= 4π I A 2 = a t Maximize a/t, but limited by local buckling and ovalization Plant stems have compliant core ( core-rind structure ) 23

24 Plant Stems: Bending Core resists ovalization and increases local buckling resistance 24

25 Plant Stems: Bending Local buckling occurs when normal stress in compressive side of cylinder equals critical stress for axisymmetric buckling under uniaxial stress Hollow cylinder: M lb = 0.939Eat 2 1 ν 2 Cylinder with compliant core: M lb = πea2 t 1 ν f δ 2 r, E core, a E shell t 25

26 Plant Stems Foam-like core can act like elastic foundation supporting outer shell, increasing local buckling moment, M lb, reducing buckling λ Hollow tube E c increasing 26

27 Plant Stems Hollow tube 27

28 Plant Stems Within the core stress decays as move radially inward, away from the shell Stresses less than 5% of maximum at a radial distance of 5λ cr Can remove inner core, leaving core thickness, c = 5λ cr 28

29 Plant Stems Species a/t Elastic foundation M lb /M eq c/λ cr Tall blue lettuce 59 Yes Oat, rye grasses 50 Yes Sedge grass, common barley 25 Yes Core increases buckling resistance for high a/t 29

30 Monocotyledon Leaves: Sandwich Structures (Also in Skulls, Cuttlefish Bone, Horseshoe Crab Shell) 30

31 Monocotyledon Leaves: Sandwich Beams Iris Bulrush 31

32 Sandwich Structures: Leaves Sclerenchyma Parenchyma 0.5 mm Iris leaf 1mm Bulrush leaf 32

33 Sandwich Structures: Leaves Stipa gigantea (Giant feather grass) Lolium perenne (Rye grass) Black = sclerenchyma White = parenchyma Vincent, 1982,

34 Monocotyledon Leaves Fibers (sclerenchyma) along outer surface of leaves Foam-like cells (parenchyma) or ribs in core Acts like structural sandwich panel Increase in moment of inertia by separating stiff faces by a lightweight core Large surface area for photosynthesis 34

35 Sandwich Beam Deflection δ = δ b + δ s = B 1 Pl 3 + ( EI) eq B 2 Pl ( AG) eq Flexural rigidity: ( EI) eq E f btc2 2 Shear rigidity: ( AG) eq bcg c Cantilever: B 1 =3 B 2 = 1 35

36 Iris Leaves t = 30 μm c = 0.5 to 3.0 mm E f = 8.2 GPa G c =2 MPa Measured stiffnesses (N/mm): Calculated stiffnesses (N/mm): Calculated/measured:

37 Conclusion Wood Uniform honeycomb increases E 1/2 /ρ, σ f 2/3 /ρ Constant ρ, E, σ f, vary r(z) in tree Palm stem Radial density gradient Constant r, vary ρ(r), E(r), σ f (r) in palm stem Increases (EI) relative to uniform distribution of solid Stress distribution across radius matches strength distribution 37

38 Conclusion Plant stems Cylindrical shell with compliant core Increases buckling resistance over equivalent hollow circular tube for large a/t Monocotyledon leaves Sandwich structure, efficient in bending Leaves provide own structural support as well as area for photosynthesis Rectangular cross-section maximizes surface area for photosynthesis 38

39 Acknowledgements Mike Ashby, Ken Easterling, Hugh Shercliff Gebran Karam, Phoebe Cheng, Ulrike Wegst, Ros Olive, Tessa Shercliff Justin Breucop, Don Galler, Beth Beighlie National Science Foundation Matoula S. Salapatas Professorship at MIT 39

40 Image References Connor S (1994) New England Natives: A celebration of people and trees. Harvard University Press. Dinwoodie J (1981)Timber: Its nature and behaviour. Van Nostrand Reinhold. Rich PM (1987) Mechanical structure of the stem of arborescent palms. Bot. Gazette 148, 42. Rich PM (1987) Developmental anatomy of the stem of Welfia Georgii, Iriartea Gigantea and other arborescent palms: Implications for mechanical support. Am. J. Botany 74,

Lecture 16-17, Sandwich Panel Notes, 3.054

Lecture 16-17, Sandwich Panel Notes, 3.054 Sandwich Panels Two stiff strong skins separated by a lightweight core Separation of skins by core increases moment of inertia, with little increase in weight Efficient for resisting bending and buckling