Frustrating geometry: Non Euclidean platesold.phys.huji.ac.il/~efrati/articles/Non Euclidean plates.pdfFrustrating geometry: Non Euclidean plates Racah Institute of Physics The Hebrew

Efi Efrati

In collaboration withEran Sharon & Raz Kupferman

September 2009

Frustrating geometry: Frustrating geometry: Non Euclidean platesNon Euclidean plates

Racah Institute of PhysicsThe Hebrew University of Jerusalem

Outline:

• Growth and geometric frustration.

• Incompatible 3D hyperelasticity.

• Non Euclidean plates: Examples.

• Reduced 2D elastic theory of non Euclidean plates.

• Non Euclidean plates: Analysis.

• An application: Almost minimal surfaces.

Outline:







Growth

Plant growth mechanics• Residual stress due to local growth.• Biological response to mechanical stress.• Manipulating growth regulation causes morphological changes.

Arabidopsis mutant Peach leaf curl CINCINNATA mutant, [Nath et al 2003]

Time scale separation

Plant growth (Arabidopsis): - Cell division hours- 2 % area growth by expansion hours

- Acoustic time 1 μs- Cantilever mode typical time 100 μs

ShapingElastic ττ <<

Spontaneous growth and geometric frustration

Why local growth is likely to result in residual stress ?

Given 8 points 28 rods of arbitrary lengths, you will (probably) not be able to connect every pair once.

“think in rod length rather than points”

Example: Isotropic non-homogeneous growth.Every point initially at rest expands isotropically by a factor .


)(rrλ


Example: Isotropic non-homogeneous growth.Every point initially at rest expands isotropically by a factor .


)(rrλ


The only expansion factor which does not result in residual stress : 2

0 ||)(

rr

Cr rrr

−=λ

Soft active deformations

Residual stress must be addressed when describing motion by auto-deformation.

Making use of unused “residual work”.

Signature of geometric frustration

Tempered glass fragments cannot be “re-joined”When tempered glass shatters, each of its fragments deforms to relax internal stresses. The relaxed fragments do not fit one another.

Signature of geometric frustration

The helicoidal form of a Bauhinia pod is residually stressed. Slicing along its length generate helices of a higher pitch.

Outline:







Hyper-elasticity

A measure of local deformation: Strain

Local elastic energy density as a function of the deformation

+

=Hyperelastic description

[Truesdell 1952]

∫∫∫= dVWE )(ε

ε∂∂

=W

SStress is obtained as a Frechet derivative of the elastic energy density.

A measure of deformation: The Strain

Stress free configuration

x

Deformed configuration

r)(xr

xru −=The displacement field

( ) ( )Irruuuu TTT −∇∇=∇∇+∇+∇=ε )()()( 21

21 Cauchy St-Venant strain

A measure of deformation: The Strain

Stress free configuration

x

Deformed configuration

r)(xr

xru −=The displacement field

Rely on the existence of a stress free configuration

Differential geometry I : Three dimensions

)(xr• The metric tensor:

• Length element:

A 3 by 3 matrix is the metric of a body in Euclidean space.

jiij dxdxgds =2

jiijx

r

x

rg

∂∂

⋅∂∂

=rr

All components of the Riemann curvature tensor vanish.

Such metrics are called:

• Euclidean

• Compatible

• Embeddable

If two embeddable metrics are identical, the bodies they describe differ by a rigid motion.

The energy stored within a deformed elastic body is a volume integral of an elastic energy density which depends only on the local metric tensor and on tensors that characterize the body, but are independent of the configuration.

3D “incompatible” hyperelasticity

3D metric determine a configuration uniquely.

Our adaptation of Truesdell’s hyperelasticityprinciple is formulated in terms of metric.

• The energy density depends on the configuration through the metric.

• For every there exists a unique SPD tensor for which the energy density vanish, i.e.

• is called the reference metric.

• The Cauchy-St. Venant strain

current configuration metric reference metric


)( xgW ,

)(xgx

0))(( =xxgW ,

)(xg( )gg −= 2

1ε

0 0)( =⇔= εε xW ,

When a stress free configuration exists, we may set .

In such coordinates the Cauchy-St. Venant strain adopts the familiar form


Ig =

( ) ( ) ( )uuuuIrrIg TTT ∇∇+∇+∇=−∇∇=−=ε )()()( 21

21

21




Lame coefficients

Ig =

( )3εεε OAW klijijkl += '

( ) ( ) ( )uuuuIrrIg TTT ∇∇+∇+∇=−∇∇=−=ε )()()( 21

21

21

For such a case the elastic energy density of an isotropic material is of the form

( )jkiljlikklijijklA δδδδμδλδ ++='




Lame coefficients

Ig =

( )3εεε OAW klijijkl += '

( ) ( ) ( )uuuuIrrIg TTT ∇∇+∇+∇=−∇∇=−=ε )()()( 21

21

21

For such a case the elastic energy density of an isotropic material is of the form


A stress free configuration may not exist.

Locally (only at a point) we may set new coordinates in which


klijijklAW ''' εε2

1=

( )ijijij g δ−=ε '' 21


Ig ='




1=



Ig ='

Tensor transformation rules (Covariance becomes valuable)

ikki xx ∂∂=Λ /'

ikki xx '/ ∂∂=)(Λ-1




1=



klijijklAW εε2

1=

( )ijijij gg −= 21ε

( )jkiljlikklijijkl ggggggA ++= μλ

Ig ='



ikki xx '/ ∂∂=)(Λ-1




1=



klijijklAW εε2

1=

( )ijijij gg −= 21ε

( )jkiljlikklijijkl ggggggA ++= μλ

Ig ='

kllj

kiijg δΛΛ=



ikki xx '/ ∂∂=)(Λ-1

Recapitulation


ki

ik

kk

iiklij

ijklAW εμεεεεε λ +== 221( )ijijij gg −= 2

1ε

321 dxdxdxgWE ∫∫∫= ||

Strain is measured with respect to a reference metric.

[Efrati Sharon & Kupferman, JMPS 2009]

Recapitulation


The reference metric need not be immersible.

ki

ik

kk

iiklij

ijklAW εμεεεεε λ +== 221( )ijijij gg −= 2

1ε

321 dxdxdxgWE ∫∫∫= ||

Strain is measured with respect to a reference metric.

[Efrati Sharon & Kupferman, JMPS 2009]

Outline:







Environmentally responsive gels .

• N-Isopropylacrylamide undergoes a large volume reduction

when heated above 33 co .

• The volume reduction ratio depends strongly on monomer

concentration.

• When injected cold with a catalyst, the gel polymerizes in

a few minutes, “freezing” the monomer concentration

gradient.

Experimental realization of Non Euclidean plates

[Klein Efrati & Sharon. Science 2007]


[Klein Efrati & Sharon. Science 2007]

Does not shrink

Shrinkage homogeneous across the thickness

Shrinks considerably

Large thickness: Flat and strained.

Small thickness: Buckled.


Does not shrink

Shrinkage homogeneous across the thickness

Shrinks considerably

Large thickness: Flat and strained.

Small thickness: Buckled.



Hyperbolic metric (saddle like everywhere)


The only shaping mechanism is the Prescription of a Non-Euclidean Metrics

Shaping by metric prescription

Crocheted segment of the hyperbolic plane [Taimina 1997].

Uncontrolled tearing

Controlled tearing

[Nath et-al 2003]

[E. Coen et-al 2003]

Leaf shaping Plastic deformation Crocheting & knitting

Computer simulations

[Seung & Nelson 1988]

[Koehl et al 2008]

Defects in flexible membranes with crystalline order

[Marder et-al 2004]

Three layers of identical array of springs of spatially varying rest length.

Recent implementation to macroalgae blades

Outline:







+

-

Differential geometry II : Surfaces

The first quadratic form (metric)

The second quadratic form (curvatures)

rra βααβ ∂⋅∂=

Nrb ˆ⋅∂= αβαβ

( ) ( )βα=κ+κ= bTrH 21

2121 ( )βα=κκ= bK det21

),,( αβδγαβγαβ ∂∂∂= aaaFKGauss’ Theorema Egregium

The mean curvature The Gaussian curvature

+

-






( ) ( )βα=κ+κ= bTrH 21

2121 ( )βα=κκ= bK det21


The mean curvature The Gaussian curvature

),,,(0 2,1 αβγαβαβγαβ ∂∂= bbaaG

Peterson Mainardi Codazzi eq.

+

-







),,,(0 2,1 αβγαβαβγαβ ∂∂= bbaaG

Peterson Mainardi Codazzi eq.

Given two quadratic forms which satisfy all three GPMC equations, they define a surface uniquely.

A toy model

Too long

Similar lengths

Similar lengths

Small thickness: Buckled configuration. The trapezoid is bent to accommodate the excess length.

Large thickness: Flat configuration. The trapezoid is under compression.

Trapezoidal inclusion

• “Rest lengths” are continuous

• The body possess no internal structure across the thin dimension

No stress free configuration

A toy model

Neither plates nor shells

A plate may be considered as a stack of identical surfaces

Short

Long

A shell may be considered as a continuous collection of non-identical surfaces

Similarly to plates: no structural variation across the thin dimension

Unlike plates: 2D metric of mid-surface non-Euclidean


A plate may be considered as a stack of identical surfaces

Short

Long

A shell may be considered as a continuous collection of non-identical surfaces

Similarly to plates: no structural variation across the thin dimension

Existing plate/shell theories do no apply

Unlike plates: 2D metric of mid-surface non-Euclidean


denotes the coordinate along the thin dimension

Non-Euclidean plate metric

23

2tt x ≤≤−

.0,10000

32221

1211

=∂⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= αβaaa

aa

g

0)(0)( 32 ≠⇔≠ DijklD gRaK

Reduced 2D elastic energy

• Integrate out the thin dimension.

• Assumption: Faces of sheet are free of forces.

[ ]∫ ∫∫−

− ⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

2

2

32121)1(2

t

t

dxdxdxgY

E kk

ii

ik

ki ||εεεε

ν νν

Goal: Describe the elastic behavior, using only mid-surface properties; First and second fundamental forms.


Goal: Describe the elastic behavior, using only mid-surface properties; First and second fundamental forms.

The first quadratic form (metric) of the mid-surface

rra βααβ

The second quadratic form (curvatures) of the mid-surface

∂⋅∂= Nrb ˆ⋅∂= αβαβ

( )γδαβννβδαγ

ναβγδ aaaaA Y

−+ += 11

tohdxdxabbAt

dxdxaaaaaAtE

..||

||))((21

2413

2181

+

+−−=

∫∫∫∫

γδαβαβγδ

γδγδαβαβαβγδ


BS etetE 3+=

Stretching content Bending content

2181 ||))(( dxdxaaaaaAeS ∫∫ γδγδαβαβ

αβγδ −−=

21241 || dxdxabbAeB ∫∫ γδαβ

αβγδ=

Generalization of F.V.K and Koiter theories for large displacements and arbitrary intrinsic geometry.

Bending content measures the magnitude of curvatures.

Vanishes only for flat configuration .

Stretching content measures deviations from the given 2D metric.

Vanishes only if the midsurface is an isometric embedding of .a


BS etetE 3+=

Stretching content Bending content

2181 ||))(( dxdxaaaaaAeS ∫∫ γδγδαβαβ

αβγδ −−=

21241 || dxdxabbAeB ∫∫ γδαβ

αβγδ=

Small thickness:

• Stretching term dominates

• 2D isometric solution

Large thickness:

• Bending term dominates

• Flat solutions (unbuckled)

Note: and are not independent variables.αβbαβa

[Efrati Shron & Kupferman JMPS 2009]


[Cerda & Mahadevan 2003-4][Huang et-al 2007]

Current literature: Elastic substrate, gravitational force, capillary forces, tension, confinement or some other mechanism necessitates buckling. Competition between the two energy terms yields the buckling length scales.

[Vella et-al 2009]


[Cerda & Mahadevan 2003-4][Huang et-al 2007]

Current literature: Elastic substrate, gravitational force, capillary forces, tension, confinement or some other mechanism necessitates buckling. Competition between the two energy terms yields the buckling length scales.

No external force; Confinement is replaced by an embedding problem.

[Efrati Klein & Sharon. Physica D 2007]

[Vella et-al 2009]

Outline:






- Buckling

- The vanishing thickness limit

- Equipartition


Hemispherical plate

We numerically minimize the elastic energyBS etteE 3+=

We use a reference metric of a spherical cap

The solution is assumed to be axially symmetric

⎟⎟⎠

⎞⎜⎜⎝

⎛=

)(001

12 xa

sin

1110 1 .. ≤≤ x

Hemispherical plate

BS etteE 3+=

Hemispherical plate

BS etteE 3+=

∞Ste

03Bet

Hemispherical plate

BS etteE 3+=

∞Ste

03Bet

Similar flat configurations

Almost isometric embedding

Hemispherical plate

BS etteE 3+=

∞Ste

03Bet


Almost isometric embedding

Slightly buckled

Hemispherical plate

∞Ste


0)( ≠aK

Analytic results:

• The unbuckled state must contain both compression and tension.

• If , there exists a finite buckling thickness.

• Classification in terms of the plane stress in the unbuckled configuration to super-critical and sub-critical bifurcations.

[Efrati et-al PRE 2009]

Hemispherical plate

We have proved that In the limit

i.e. an isometric embedding of mid-surface.

0→t αβαβ aa =

Hemispherical plate

We have proved that In the limit

i.e. an isometric embedding of mid-surface.

0→t αβαβ aa =

The isometry is (usually) not unique.

The limit configuration will minimize the bending content amongst all isometries.

Recent Gamma limit proof for non Euclidean plates directly from 3D [Lewicka & Pakzad 2009].

Hemispherical plate

BS etteE 3+=

27t∝

03Bet

00⎯⎯→⎯→tSe

00B

tB ee ⎯⎯→⎯→

No mutual zero to the two terms; No equipartition.

Hemispherical plate

( ) 2703 teet BB ∝−

The variation of the bending energy about its limit

BS ete && 2−=

BS etteE 3+=

00⎯⎯→⎯→tSe

00B

tB ee ⎯⎯→⎯→

Hemispherical plate

( ) 2703 teet BB ∝−

The variation of the bending energy about its limit

The power 7/2 comes from a boundary layer whose thickness scales as

||.

k

tl LB ∝

[Efrati et-al PRE 2009]

Outline:







Solving the vanishing thickness limit

Gaussian curvature Mean curvature

Provided there exists an isometric embedding with bounded energycontent, the vanishing limit configuration minimizes the Willmoreenergy among all isometric embeddings.

( ) ( )dAKHdAkkkk ∫∫∫∫ ν−−=ν++ 212

2122

214

1 2

21kkK = ( )2121 kkH +=

Solving the vanishing thickness limit

Gaussian curvature Mean curvature

[Willmore 1986]

Constrained minimization problem: The GMPC equations.

Provided there exists an isometric embedding with bounded energycontent, the vanishing limit configuration minimizes the Willmoreenergy among all isometric embeddings.

dAH∫∫ 2

21kkK = ( )2121 kkH +=

Differential geometry III: Isometric embedding of surfaces in 3D

The Gauss Peterson Mainardi Codazzi compatability conditions may be considered as evolution equations for the curvature form, .

• For elliptic set of P.D.Es .

• For hyperbolic set of P.D.Es .

K<0

0<K

αβb

2

91 −= cmK

2

91 −−= cmK

cmt 06.0=cmt 075.0= cmt 019.0=cmt 025.0=

Almost minimal surfaces

Minimal surfaces are surfaces of vanishing mean curvature

• Constitute trivial minima of the Willmore energy.

• Not every metric can be embedded as a minimal surface.

H = 0

dAH∫∫ 2


Minimal surfaces are surfaces of vanishing mean curvature

• Constitute trivial minima of the Willmore energy.

• Not every metric can be embedded as a minimal surface.

H = 0

The mean curvature of any hyperbolic metric can vanish along a curve

dAH∫∫ 2


We considering a thin narrow hyperbolic strip Lwt <<<<

dAHE

w

w

W ∫ ∫−

=2

2

2

w

L


We considering a thin narrow hyperbolic strip Lwt <<<<

( )dxweweweeEW ......33

2210 ++++= ∫

w

L


We considering a thin narrow hyperbolic strip

• On the mid-line of the strip the mean curvature vanishes.

• The resulting ribbons resemble minimal surfaces.

• Both chiralities possess the same energy, thus are equally probable.

Lwt <<<<

)( 5wOEW =

Almost minimal surfacesInspired by the result that every (narrow enough) hyperbolic metric can be embedded as an almost minimal surface, we found new family of pseudospherical embeddings (surfaces of constant Gaussian curvature: K=-1).

Almost minimal surfacesInspired by the result that every (narrow enough) hyperbolic metric can be embedded as an almost minimal surface, we found new family of pseudospherical embeddings (surfaces of constant Gaussian curvature: K=-1).

The new family of pseudospherical embeddings also admits embeddings of very large mean curvature.


Shrinks

Do not Shrink

Prescribing a hyperbolic geometry on a strip using responsive gels


Shrinks

Do not Shrink

Prescribing a hyperbolic geometry on a strip using responsive gels

• The mean curvature quadratic in the width coordinate.


0.0 0.4 0.8 1.2 1.6 2.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

CenterEdge H2 /K

Cur

vatu

re S

quar

ed (m

m-2)

Distance from the Edge (mm)

K H2

-H2/K

Preliminary results

• The mean curvature quadratic in the width coordinate.

• Results become irrelevant for wide strips.


0.0 0.4 0.8 1.2 1.6 2.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

CenterEdge H2 /K

Cur

vatu

re S

quar

ed (m

m-2)

Distance from the Edge (mm)

K H2

-H2/K

Preliminary results

Conclusion

• Residual stress inevitable in local shaping mechanisms.

• “Incompatible” elasticity is the theoretical tool to deal with residual stress.

• Non-Euclidean plates: - Derivation from 3D elasticity.

- Buckling, Boundary layer.

- The isometric embedding problem.

• Application: Almost minimal surfaces.

Conclusion

• Residual stress inevitable in local shaping mechanisms.

• “Incompatible” elasticity is the theoretical tool to deal with residual stress.

• Non-Euclidean plates: - Derivation from 3D elasticity.

- Buckling, Boundary layer.

- The isometric embedding problem.

• Application: Almost minimal surfaces.

• Natural extension: Non-Euclidean shells.

Elastic theory of non-Euclidean plates and shells [submmited to Nonlinearity 2009]

• Geometric applications: Folding and controllability of non Euclidean plates.

The EndThe End

Frustrating geometry: Non Euclidean platesold.phys.huji.ac.il/~efrati/articles/Non Euclidean plates.pdfFrustrating geometry: Non Euclidean plates Racah Institute of Physics The Hebrew

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Frustrating geometry: Non Euclidean platesold.phys.huji.ac.il/~efrati/articles/Non Euclidean plates.pdfFrustrating geometry: Non Euclidean plates Racah Institute of Physics The Hebrew