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Mechanically Efficient Cellular Microstructures in Plants

Lorna J. GibsonMaterials Science & Engineering

MIT

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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

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Introduction

• Wood: Uniform honeycomb-like structure

• Palm stem: Radial density gradient• Plant stem: Cylindrical shell with

compliant core• Monocotyledon leaves: Sandwich

structures

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Wood:Honeycomb-Like Microstructure

Cedar

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Wood: Honeycomb Models

Cell wall:FiberCompositeModelE*

Esalong

=ρ*

ρs

E*

Es across

=ρ*

ρs

⎛

⎝⎜⎞

⎠⎟

3

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Wood in Bending: E1/2/ρ E*( )1/2

ρ* =Es( )1/2

ρs

ρs

ρ*

⎛⎝⎜

⎞⎠⎟

1/2

Stiffness performance index for wood inbending is similar to that for bestengineering composites

Wood cell wall

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Wood in Bending: σf2/3/ρ

σ f*( )2 /3

ρ* =σ ys( )2 /3

ρs

ρs

ρ*

⎛⎝⎜

⎞⎠⎟

1/3

Strength performance index for wood inbending is similar to that for best engng composites

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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)

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Palm Stem: Radial Density Gradient

(Also Bamboo)

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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

http://en.wikipedia.org/wiki/Image:Palmtree_Curacao.jpg

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Palm: Microstructure ofPeripheral Stem Tissue

Young Old

6 μm 10 μm

Rich, 1987

Kuo-Huang et al., 2004

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Palm Stem: Density Gradient

Rich, PM (1987) Bot.Gazette 148, 42-50.

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Palm Stem: Density at Breast Height

A single mature palm has a similar range of density as nearly all species of wood combined

Rich, PM (1987) Bot.Gazette 148, 42-50.

Densitiesof commonwoods

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Palm Stem: Density Gradient

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Palm Stem: Mechanical Properties vs. Density

Rich, PM (1987) Bot.Gazette 148, 42-50.

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Density Gradient:Iriartea gigantea

ρ =rro

⎛⎝⎜

⎞⎠⎟

n

ρmax

E = Cρ

ρmax

⎛

⎝⎜⎞

⎠⎟

m

= Crro

⎛

⎝⎜⎞

⎠⎟

mn

EI( )gradient =Cπro

4

mn + 4

EI( )gradient

EI( )uniform

=4

mn + 4n + 2

2⎛⎝⎜

⎞⎠⎟

m

Iriartea palm: n = 2, m = 2.5, (EI)gradient/(EI)uniform = 2.5

Similar calculation for Welfia georgii, gives(EI)gradient/(EI)uniform = 1.6

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Palm Stem:Bending Stress Distribution

σ (y) = Eε = Eκ y

σ (r,θ ) = Crro

⎛

⎝⎜⎞

⎠⎟

mn

κ r cosθ ∝ rmn+1

Iriartea gigantea: m = 2.5, n =2

σ ∝ r6

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Palm Stem:Bending Strength Distribution

σ * ∝ρ

ρmax

⎛⎝⎜

⎞⎠⎟

q

∝rro

⎛⎝⎜

⎞⎠⎟

nq

Iriartea gigantea: n = 2, q = 2

σ * ∝ r4

Strength matchesbending stress distribution

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Plant Stems:Cylindrical Shells with

Compliant Cores

(Also in Animal Quills,Toucan Beak)

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Plant Stems

Milkweed Grassy stem

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Milkweed Stem

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Grassy Stem

Hollow struts

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Plant stems

• Circular tube cross-section• Resists bending (wind loads)• Maximize shape factor

• Maximize a/t, but limited by local buckling and ovalization

• Plant stems have compliant core (“core-rind structure”)

Φ =4π IA2 =

at

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Plant Stems: Bending

• Core resists ovalization and increases local buckling resistance

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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:

• Cylinder with compliant core:

Mlb =0.939Eat 2

1− ν 2

Mlb =πEa2t1− ν 2

fδr

,Ecore

Eshell

,at

⎧⎨⎩

⎫⎬⎭

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Plant Stems• Foam-like core can act like elastic foundation supporting outer

shell, increasing local buckling moment, Mlb, reducing buckling λ

Hollow tube

Ec increasing

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Plant Stems

Hollow tube

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Plant Stems

• Within the core stress decays as move radially inward, away from the shell

• Stresses less than 5% of maximum at aradial distance of 5λcr

• Can remove inner core, leaving core thickness, c = 5λcr

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Plant Stems

25

50

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a/t

3.270.77YesSedge grass, common barley

3.811.26YesOat, rye grasses

3.811.37YesTall blue lettuce

c/λcrMlb/MeqElastic foundation

Species

Core increases buckling resistance for high a/t

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Monocotyledon Leaves:Sandwich Structures

(Also in Skulls, Cuttlefish Bone, Horseshoe Crab Shell)

31Iris Bulrush

Monocotyledon Leaves: Sandwich Beams

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Sandwich Structures: Leaves

Iris leaf

Bulrush leaf1mm

0.5 mm

SclerenchymaParenchyma

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Sandwich Structures: Leaves

Lolium perenne(Rye grass)

Stipa gigantea(Giant feather

grass)

Vincent, 1982,1991

Black = sclerenchymaWhite = parenchyma

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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

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Sandwich Beam Deflection

δ = δb + δ s =Pl 3

B1 EI( )eq

+Pl

B2 AG( )eq

EI( )eq ≈Ef btc2

2AG( )eq ≈ bcGcFlexural rigidity: Shear rigidity:

Cantilever: B1 = 3 B2 = 1

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Iris Leaves

t = 30 μm

c = 0.5 to 3.0 mm

Ef = 8.2 GPa

Gc = 2 MPa

Measured stiffnesses (N/mm): 0.66 0.54 0.41 0.25Calculated stiffnesses (N/mm): 1.21 0.78 0.51 0.29

Calculated/measured: 1.83 1.44 1.24 1.16

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Conclusion

• Wood– Uniform honeycomb increases E1/2/ρ, σf

2/3/ρ– Constant ρ, E, σf, vary r(z) in tree

• Palm stem– Radial density gradient– Constant r, vary ρ(r), E(r), σf(r) in palm stem– Increases (EI) relative to uniform distribution of

solid– Stress distribution across radius matches strength

distribution

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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

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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

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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, 792.